Best Approach
Permutation & Combination
(Sheet)
By Mathematics Wizard
Manoj Chauhan Sir (IIT Delhi)
No. 1 Faculty of Unacademy,
Exp. More than 13 Years in
Top Most Coachings of Kota
Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
KEY CONCEPTS
DEFINITIONS :
1.
PERMUTATION : Each of the arrangements in a definite order which can be made by taking some or
all of a number of things is called a PERMUTATION.
2.
COMBINATION : Each of the groups or selections which can be made by taking some or all of a
number of things without reference to the order of the things in each group is called a COMBINATION.
FUNDAMENTAL PRINCIPLE OF COUNTING :
If an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways,
then the total number of different ways of simultaneous occurrence of both events in a definite order is
m × n. This can be extended to any number of events.
RESULTS :
(i)
A Useful Notation : n! = n (n 1) (n 2)......... 3. 2. 1 ; n ! = n. (n 1) !
0! = 1! = 1 ; (2n)! = 2n. n ! [1. 3. 5. 7...(2n 1)]
Note that factorials of negative integers are not defined.
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
If nP r denotes the number of permutations of n different things, taking r at a time, then
n!
nP = n (n 1) (n 2)..... (n r + 1) =
Note that , nPn = n !.
r
( n r )!
If nCr denotes the number of combinations of n different things taken r at a time, then
n
n!
Pr
nC =
=
where r n ; n N and r W.
r
r!(n r )!
r!
The number of ways in which (m + n) different things can be divided into two groups containing m & n
(m n ) !
things respectively is :
If m = n, the groups are equal & in this case the number of subdivision
m!n!
( 2n )!
is
; for in any one way it is possible to interchange the two groups without obtaining a new
n! n!2!
distribution. However, if 2n things are to be divided equally between two persons then the number of
(2n )!
ways =
.
n!n!
Number of ways in which (m + n + p) different things can be divided into three groups containing m , n
( m n p )!
& p things respectively is
, m n p.
m! n! p!
(3n )!
If m = n = p then the number of groups =
.
n!n!n!3!
(3n )!
However, if 3n things are to be divided equally among three people then the number of ways =
.
( n!) 3
The number of permutations of n things taken all at a time when p of them are similar & of one type, q of
them are similar & of another type, r of them are similar & of a third type & the remaining
n – (p + q + r) are all different is : n! .
p!q!r!
The number of circular permutations of n different things taken all at a time is ; (n 1)!. If clockwise &
anticlockwise circular permutations are considered to be same, then it is
( n 1)!
.
2
Note : Number of circular permutations of n things when p alike and the rest different taken all at a time
(n 1)!
distinguishing clockwise and anticlockwise arrangement is
.
p!
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
(viii)
Given n different objects, the number of ways of selecting atleast one of them is ,
nC + nC + nC +.....+ nC = 2n 1. This can also be stated as the total number of combinations of n
1
2
3
n
distinct things.
(ix)
Total number of ways in which it is possible to make a selection by taking some or all out of
p + q + r +...... things , where p are alike of one kind, q alike of a second kind , r alike of third kind &
so on is given by :
(p + 1) (q + 1) (r + 1)........ –1.
(x)
Number of ways in which it is possible to make a selection of m + n + p = N things , where p are alike
of one kind , m alike of second kind & n alike of third kind taken r at a time is given by coefficient of xr
in the expansion of
(1 + x + x2 +...... + xp) (1 + x + x2 +...... + xm) (1 + x + x2 +...... + xn).
Note : Remember that coefficient of xr in (1 x)n = n+r1Cr (n N). For example the number of ways
in which a selection of four letters can be made from the letters of the word PROPORTION is given by
coefficient of x4 in (1 + x + x2 + x3) (1 + x + x2) (1 + x + x2) (1 + x) (1 + x) (1 + x).
(xi)
Number of ways in which n distinct things can be distributed to p persons if there is no restriction to the
number of things received by men = pn.
(xii)
Number of ways in which n identical things may be distributed among p persons if each person may
receive none , one or more things is ; n+p1Cn.
(xiii)
a.
nC
r
= nCnr ; nC0 = nCn = 1
c.
nC
r
+ nCr1 =
n+1C
;
b.
nC
x
= nCy x = y or x + y = n
r
n
n 1
n 1
if n is even. (b) r =
or
if n is odd.
2
2
2
(xiv)
nC
(xv)
Let N = pa. qb. rc...... where p , q , r...... are distinct primes & a , b , c..... are natural numbers then:
(a)
The total numbers of divisors of N including 1 & N is = (a + 1)(b + 1)(c + 1).....
r is maximum if : (a) r =
(b)
The sum of these divisors is
= (p0 + p1 + p2 +.... + pa) (q0 + q1 + q2 +.... + qb) (r0 + r1 + r2 +.... + rc)....
(c)
Number of ways in which N can be resolved as a product of two
factors is =
1 (a 1)(b 1)(c 1)....
2
1
2
(a 1)(b 1)(c 1).... 1
if N is not a perfect square
if N is a perfect square
Number of ways in which a composite number N can be resolved into two factors which are
relatively prime (or coprime) to each other is equal to 2n1 where n is the number of different
prime factors in N.
Grid Problems and tree diagrams.
(d)
(xvi)
DEARRANGEMENT :
Number of ways in which n letters can be placed in n directed letters so that no letter goes into its own
1 1 1
n 1
envelope is = n! ...........(1)
.
n!
2! 3! 4 !
(xvii) Some times students find it difficult to decide whether a problem is on permutation or combination or
both. Based on certain words / phrases occuring in the problem we can fairly decide its nature as per the
following table :
PROBLEMS OF COMBINATIONS PROBLEMS OF PERMUTATIONS
Selections , choose
Distributed group is formed
Committee
Geometrical problems
Arrangements
Standing in a line seated in a row
problems on digits
Problems on letters from a word
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
SOLVED EXAMPLES
The maximum number of points into which 4 circles and 4 straight lines intersect is
1.
(A) 26
Sol.
2.
(D) 72
4P
4 lines intersect each other in
6 points 4 circles intersect in 2 = 12 points.
Also, each line cuts 4 circles into 8 maximum points
4 lines cut four circles into 32 points.
Hence, Required number = 6 + 12 + 32 = 50
A candidate is required to answer 6 out of 10 questions which are divided into parts containing 5 questions each. He
is permitted to attempt not more than 4 from any group. The number of ways in which he can make up his choice is
(B) 100
(C) 220
(D) 240
Number of Questions = 10
Number of Questions to Answer = 6
Max number of Questions to Answer from each group = 4
___________________________________________________
Section A(5)
Section B (5)
___________________________________________________
Case - I
2
4
Case - II
3
3
Case
II
4
2
___________________________________________________
So the number of ways he can make up his choice are
5C · 5C + 5C 5C + 5C 5C
2
4
3
3
4
2
Required ways = 50 + 100 + 50 = 200
The number of rectangles in the following figure is
3.
(A) 5 × 5
Sol.
4.
Sol.
5.
(C) 5C2 × 5C2
(D) None of these
(B) 638
(C) 639
(D) 640
Let number of ways of voting = W
W = 10C1 + 10C2 + 10C3 + 10C4 + 10C5
W = 10 + 45 + 120 + 210 + 252
W = 637
If in a chess tournament each contestant plays once against each of the others and in all 45 games are played, then the
number of participants is
(A) 9
Sol.
(B) 5P2 × 5P2
Since, there are 5 horizontal lines and 5 vertical lines, and each choice of a pair of horizontal lines and a pair of vertical
lines gives us a rectangle. Hence the number of rectangles = 5C2 × 5C2.
In an election a man can vote for any number of candidates but he can not vote for more than which are to be elected.
There are 10 candidates and five are to elected. The number of ways of voting by a man is
(A) 637
(B) 10
(C) 15
(D) None of these
Let there be n participants
nC = 45
2
6.
(C) 56
4C =
2
(A) 200
Sol.
(B) 50
n(n 1)
45
2
n2 – s – 9 = 0
n = 10
The number of ways in which 13 gold coins can be distributed among three persons such that each one gets at least
2 gold coins is
(A) 36
(B) 24
(C) 12
(D) 6
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Sol.
Permutation & Combination
Since, each person must get at least 2 gold coins. Let us first give 2 gold coins to each of 3 persons.
Initially
13 Coins
Distribute 2 coins to each of them
Finally
7 Coins
x1 + x2 + x3 = 7 required ways = 9C2 = 36
The number of ways of chossing 10 balls from infinite white, red, blue and green balls is
7.
(A) 70
(B) 84
(C) 286
(D) 86
Sol.
Let x1, x2, x3, x4 are the number of balls chosen from infinite white, red, blue and green balls respectively
x1 + x2 + x3 + x4 = 10
required based = 13C3 = 286
8.
Give 5 different green dyes, 4 different blue dyes and 3 different red dyes, the number of combinations of dyes thatcan
be chosen by taking atleast one green and one blue dye is
(A) 248
(B) 120
(C) 3720
(D) 465
Sol.
Number of ways of selecting at least one green dye out of 5 different green dyes
5C + 4C + 5C + 5C + 5C = 25 – 1
1
2
3
4
5
Also, at least one blue dye can be selected out of 4 blue dyes in
4C + 4C + 4C + 4C + 4C = 24 – 1
1
2
3
4
5
Again, 3 different red dyes can be selected in
3C + 3C + 3C + 3C + 3C = 23 – 1
1
2
3
4
5
Required ways = (25 – 1) (24 – 1) (23) = 3720
9.
20 persons were invited for a party. The different number of ways in which they can be seated on a circular table with
two with particular persons seated on either side of host is
(A) 19!
(B) 18!
(C) 20! × 2
(D) 18! × 2
Sol.
Lets group 2 particular persons along either side of host.
Now 18 persons + 1 Group can be arranged around round table in (19 – 1)! ways.
Also, 2 particular persons on either side of host can be seated in 2! ways
Total number of ways = 18!.2!
10.
There are 4 parcels and 5 post offices. In how many ways can 4 parcels be got registered ?
(A) 20
(B) 45
(C) 54
Sol.
1st parcel can be registered in 5 ways
3rd parcel can be registered in 5 ways
11.
How many words can be formed by taking 4 letters at a time out of letters of the word PROPORTION?
(A) 700
(B) 750
;
;
(D) 54 – 45
(C) 758
2nd parcel can be registered in 5 ways
4th parcel can be registered in 5 ways
(D) 800
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Sol.
Permutation & Combination
In proportion, total number of alphabets are 10. In which there are 6 distinct alphabets.
(P P) (R R) (O O O) T I N
Case I : All different
PROTIN
Required ways = 6C4 × 4! = 360
Case II : 2 Alike of one kind and 2 different
(P P) or (R R) or (O O) T I N
Out of either 2 P's or 2R's or 2 O's one pair can be selected in 3C1 ways.
Now, out of the other two distinct alphabets remain unselected and the other 3 distinct alphabets (total of 5) two can
be selected in 5C2 ways.
Hence, the total number of ways to arrange a 4 character word in this case is 3C1 × 5C2 ×
4!
360
2!
Case III : 2 Alike of one kind and 2 alike of another
(P P) (R R) (T T)
Out of 3 pairs, two can be selected in 3C2 ways.
Number of ways to arrange the alphabets
3
= C2 .
4!
18
2!2!
Case IV : 3 Alike of one kind and 1 different
(O O O) P R T I N
Since, there are only three alike O's
the single triplet can be selected in only one way.
Now, out of 5 distinct alphabets one can be selected in 5C1 ways
5
Required ways = C1
4!
20
3!
Now, the total number of ways of forming a four letter word out of the letter of word PROPORTION is :
CASE I + II + III + IV + V
Required ways = 360 + 360 + 18 + 20 = 758
12.
All the possible words obtained by rearranging the letters of the word RACE are arranged as in a dictionary.
The rank of the word CARE is
(A) 6
(B) 7
(C) 8
(D) 24
Sol.
A
C
E
R (alphabetical order)
A
_
_
_
= 3!
C
A
E
R
=1
C
A
R
E
=1
_____________________________
Rank of word care is 8
13.
The number of ways of arranging 5 players to throw the cricket ball so that the youngest may not throw first is
(A) 119
(B) 96
(C) 24
(D) 23
Sol.
Keeping the youngest player aside, one of four players can throw the cricket ball at first place in 4C1 ways.
Now, the 3 players (not able to throw) the ball in first place) and 1 youngest player i.e. 4 can arrange themselves in 4!
ways to throw the ball.
Required ways = 4C1 4! = 96
14.
Four digit numbers are formed by using the digits 0, 1, 2, 3, 4, 5, 6 (no digit is used more than once in each number). The
number of numbers divisible by 5 is
(A) 240
(B) 210
(C) 220
(D) 200
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Sol.
Permutation & Combination
The number will be divisible by 5, if there is 0 or 5 at the last of each number
0
5
Case I : Zero at last place
1, 2, 3, 4, 5, 6
Zero can be fixed in units place in 1 way only.
Since, repetition is not allowed,
for 3 places, 3 digits out of 6 can be selected in 6C3 ways and arranged in 3! ways.
Required ways = 1 · 6C3 · 3! = 120
.....(i)
Case II : 5 at last place
5 can be fixed in units place in 1 way
Now, 0 cannot occur in first place so out of 5 digits (1, 2, 3, 4, 6) first place can be filled in 5C1 ways.
Now, the 4 digits (not able to occupy the first place) and along with ZERO (0) i.e. 5 can fill the left two places in 5C2
ways and these can be arranged in 2! ways.
Required ways = 1 · 5C1 5C2 2! = 100
......(ii)
Hence, total number of Required ways = 120 + 100 = 220
15.
ˆ where a, b, c {1, 2, 3, 4, 5} such that 2a + 3b + 5c is divisible by
Number of points having position vector aiˆ bjˆ ck,
4 is
(A) 140
Sol.
(B) 70
(C) 100
(D) None of these
4m = 2a + 3b + 5c = 2a + (4 – 1)b + (4 + 1)c
4m = 4k + 2a + (–1)b + (1)c
Required number = 1 × 2 × 5 + 4 × 3 × 5 = 70
16.
The number of triangles that can be formed by joining the angular points of decagon is :
(A) 120
(B) 720
(C) 80
(D) 1000
Sol.
From 10 non-collinear angular points of a decagon, the number of valid combination of 3 points are the number of
triangles formed
10C
i.e.
3
17.
An examination paper, which is divided into two groups consisting of 3 and 4 questions respectively carries the note.
It is not necessary to answer all the questions. One question atleast should be answered from each group. THe
number of ways can an examinee select the questions is :
(A) 22
(B) 105
(C) 3P3 × 4P4
(D) 3C3 × 4C4
Sol.
Number of ways at least one question can be selected out of 3 are
3C + 3C + 3C = 23 – 1 = 7
1
2
3
Number of ways at least one question can be selected out of 4 are
4C + 4C + 4C + 4C = 24 – 1 = 15
1
2
3
4
Total number of ways = 7 × 15 = 105
18.
In how many ways the letters of the word PERSON can be placed in the squares of the adjoining figure so that no row
remains empty?
(A) 18720
(B) 18700
(C) 8!
(D) 6!
Sol.
In PERSON total letters = 6 which are to be filled in 8 squares 6 numbers of ways of choosing 6 letters to fill in 8 squares
= 8C6 – 2 = 25 – 2 = 26
19.
A man has 7 relatives, 4 of them are ladies and 3 genetlemen. His wife has 7 relatives and 3 of them are ladies and 4
gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 genetlemen so that there are 3 of man's
relative and 3 of wife's relatives.
(A) 480
(B) 485
(C) 400
(D) 425
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Sol.
s
Required Ways
= 3C0 4C3 4C3 3C0 + 3C1 4C2 4C2 3C1 + 3C2 4C1 4C1 3C2 + 3C3 4C0 4C0 3C3
Required Ways = 485
20.
Permutation & Combination
Let y be an element of the set A {1, 2, 3, 4, 6, 10, 15, 30} and x1, x2, x3 be integers such that x1x2x3 = y, then the number
of positive integral solutions x1x2x3 = y is :
(A) 64
(B) 27
(C) 81
(D) None of these
Sol.
Number of solutions of the given equation is the same as the number of solutions of the equation
x1x2x3x4 = 30 = 2 × 3 × 5
Here x4 is there because if x1x2x3 = 15, then x4 = 2 and if x1x2x3 = 5, then x4 = 6 etc.
x4 is in fact a dummy variable.
Thus, x1x2x3x4 = 2 × 3 × 5
Each of 2, 3 and 5 will be a factor of exactly one of x1, x2, x3, x4 in 4 ways.
Required number 43 = 64
21.
A person goes in for an examination in which there are four papers with maximum of m marks from each paper. The
number of ways in which one can get 2m marks is:
(A) 2m + 1
(C)
Sol.
(B)
1
(m + 1) (2m2 + 4m + 3)
3
1
(m 1)(2m2 m 1)
3
(D) None of these
2m
Coefficient of x in
Required number = 0
1
m 4
(x x ....... x )
4
1 x m1
1 x
Co-efficient of x2m in (1 – xm+1)4 (1 – x)–4
Co-efficient of x2m in (1 – 4xm+1 + 6x2m+2 + .....) (1 – x)–4
Coefficient of x2m = 2m + 3C2m – 4. m+2Cm–1
Coefficient of x2m =
(2m 1)(2m 2)(2m 3) 4m(m 1)(m 2)
6
6
Coefficient of x2m =
(m 1)(2m 2 4m 3)
3
22.
Co-efficient of x2m in
Out of 7 gentlemen and 4 ladies a committee of 5 is to be formed. The number of ways in which this can be done so as
to include atleast 3 ladies is:
(A) 84
Sol.
(B) 90
(C) 85
(D) 91
The following CASES justify creating a committee of 5 so as to include at least 3 ladies.
Ladies (4)
Gentlemen (7)
Case I
3
2
Case II
4
1
Case III
5
0
Let Required Ways = W
W = 4C3 · 7C2 + 4C4 · 7C1
W = 4.21 + 1.7
W = 91
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
23.
Permutation & Combination
A father with 8 children takes 3 at a time to Indira park, as often as he can without taking the same three children
together more than once. How often will each child go?
(A) 21
(B) 42
(C) 336
(D) 56
Sol.
Let us assume that the father takes Child No. 1 to the Indra park everytime he goes to the park.
Now, out of seven children, the number of ways he can select two children are 7C2 = 21.
24.
How many numbers greater than a million can be formed with the digits 2, 3, 0, 3, 4, 2 and 3 if repetition of digits is not
allowecd?
Sol.
Any number greater than a million must be of 7 or more than 7 digits. Here number of given digits is seven, therefore
we have to form numbers of seven digits only.
Now there are seven digits out of which 3 occurs thrice and 2 occurs twice.
7!
420
Total number of arrangements of seven digits =
2!3!
But this also includes those arrangements of seven digits whose first digit is zero and so in fact, they give only six digit
numbers.
Number of arrangments of seven digits having zero in the first place = 1
6!
60
3!2!
0
6!
ways
3!2!
Hence required number = 420 – 60 = 360
Three married couples are to be seated in a row having six seats in cinema hall. If spouses are to be seated next to each
other, in how many ways can they be seated? Find also the number of ways of their seating if all the ladies sit together.
1 way
25.
Sol.
26.
Sol.
Let the three husbands be denoted by H1, H2, H3 and their wives be denoted by W1, W2, W3 respectively.
According to question, H1 and W1 as one person H2, W2 as one person and H3, W3 as one person, we have only 3
persons.
These 3 persons can be arranged in row in 3! ways.
But each couple can be arranged among themselves in 2! ways.
Required number = 3! 2! 2! 2! = 48
Second part : Regarding all the three ladies as one person, we have only 4 persons. These 4 persons can be arranged
in 4! ways. But 3 ladies can be arranged among themselves in 3! ways.
Required number = 4! 3! = 24 × 6 = 144
In how many ways can 7 I.A. and 5 I.Sc. Students be seated in a row so that no two of the I.Sc. students may sit
together?
Here, there is no restriction on I.A. students, therefore, first we must arrange the 7 I.A. students
× I.A. × I.A. × I.A. × I.A. × I.A. × I.A. × I.A. ×
Now 7 I.A. students can be seated in a row in 7! ways.
Now if I. Sc. students sit at the places (including the two ends) indicated by 'x'. then no two of the five I.Sc. students
will sit together.
Now there are 8 places for 5 I.SC. students.
The five I.Sc. students can be seated in 8P5 ways.
27.
Required number of ways in which 7 I.A. students and 5 I.Sc. students can sit = 8P5 × 7!
8!
7!
3!
How many ways can the letters of the word 'ARRANGE' be arranged so that
(i) the two R's are never together
(ii) the two A's are together but not the two R's
(iii) neither the two A's nor the two R's are together
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Sol.
Permutation & Combination
Total number of letters = 7
A occurs twice and R occurs twice
(i)
Total number of arrangements when there is no restriction =
Number of arrangments when two R's are together =
(ii)
7!
1260
2!2!
6!2!
360 (Considering two R's as one letter)
2!2!
Required number = 1260 – 360 = 900
Number of arrangments when the two A's are togehter =
6!2!
360
2!2!
Number of arrangements when the two A's are together and the two R's are together =
28.
Sol.
5!2!2!
120
2!2!
(Considering two A's as one letter and two R's as one letter)
Number of arrangements when the two A's are together but not the two R's = 360 – 120 = 240
(iii) Number of arrangements when the two R's are not together = 1260 – 360 = 900
Number of arrangements when the two R's are not together and the two A's are together = 240.
Required number = 900 – 240 = 660
Find the number of words which can be made using all the letters of the word 'AGAIN'. If these words are written as
in a dictionary, what will be the fiftieth word?
Alphabetical order of letters is A, A, G, I, N.
Total number of letters = 5
Number of A's = 2
Number of words beginning with = A = 1 × 4! = 24
Number of words beginning with G = 1 ×
Number of words beginning with I = 1 ×
4!
12
2!
4!
12
2!
Next word will begin with N.
First word beginning with N is NAAGI
Second word beginning with N is NAAIG
Hence 50th word = NAAIG
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-1
ELEMENTARY PROBLEMS ON PERMUTATION & COMBINATION
NOTE : USE FUNDAMENTAL PRINCIPLE OF COUNTING & ENJOY DOING THE FOLLOWING.
Q.1
In how many ways can clean & clouded (overcast) days occur in a week assuming that an entire
day is either clean or clouded.
Q.2
Four visitors A, B, C & D arrive at a town which has 5 hotels. In how many ways can they
disperse themselves among 5 hotels, if 4 hotels are used to accommodate them.
Q.3
If the letters of the word “VARUN” are written in all possible ways and then are arranged as in a
dictionary, then the rank of the word VARUN is :
(A) 98
(B) 99
(C) 100
(D) 101
Q.4
How many natural numbers are their from 1 to 1000 which have none of their digits repeated.
Q.5
A man has 3 jackets, 10 shirts, and 5 pairs of slacks. If an outfit consists of a jacket, a shirt, and a pair
of slacks, how many different outfits can the man make?
Q.6
There are 6 roads between A & B and 4 roads between B & C.
(i)
In how many ways can one drive from A to C by way of B?
(ii)
In how many ways can one drive from A to C and back to A, passing through B on both trips ?
(iii)
In how many ways can one drive the circular trip described in (ii) without using the same road more than
once.
Q.7
(i)
(ii)
Q.8
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Q.9
(i)
(ii)
(iii)
(iv)
(v)
How many car number plates can be made if each plate contains 2 different letters of English
alphabet, followed by 3 different digits.
Solve the problem, if the first digit cannot be 0. (Do not simplify)
Find the number of four letter word that can be formed from the letters of the word HISTORY.
(each letter to be used at most once)
How many of them contain only consonants?
How many of them begin & end in a consonant?
How many of them begin with a vowel?
How many contain the letters Y?
How many begin with T & end in a vowel?
How many begin with T & also contain S?
How many contain both vowels?
If repetitions are not permitted
How many 3 digit numbers can be formed from the six digits 2, 3, 5, 6, 7 & 9 ?
How many of these are less than 400 ?
How many are even ?
How many are odd ?
How many are multiples of 5 ?
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
Q.10 In how many ways can 5 letters be mailed if there are 3 different mailboxes available if each letter can be
mailed in any mailbox.
Q.11
Every telephone number consists of 7 digits. How many telephone numbers are there which do not
include any other digits but 2 , 3 , 5 & 7 ?
Q.12 (a)
(b)
In how many ways can four passengers be accommodate in three railway carriages, if each
carriage can accommodate any number of passengers.
In how many ways four persons can be accommodated in 3 different chairs if each person can
occupy only one chair.
Q.13 How many of the arrangements of the letter of the word “LOGARITHM” begin with a vowel and end
with a consonant?
Q.14 Number of natural numbers between 100 and 1000 such that at least one of their digits is 7, is
(A) 225
(B) 243
(C) 252
(D) none
Q.15 How many four digit numbers are there which are divisible by 2 .
Q.16 In a telephone system four different letter P, R, S, T and the four digits 3, 5, 7, 8 are used. Find the
maximum number of “telephone numbers” the system can have if each consists of a letter followed by a
four-digit number in which the digit may be repeated.
Q.17 Find the number of 5 lettered palindromes which can be formed using the letters from the English
alphabets.
Q.18 Number of ways in which 7 different colours in a rainbow can be arranged if green is always in the
middle.
Q.19 Two cards are drawn one at a time & without replacement from a pack of 52 cards. Determine the
number of ways in which the two cards can be drawn in a definite order.
Q.20 Numbers of words which can be formed using all the letters of the word "AKSHI", if each word begins
with vowel or terminates in vowel .
Q.21 A letter lock consists of three rings each marked with 10 different letters. Find the number of ways in
which it is possible to make an unsuccessful attempts to open the lock.
Q.22 How many 10 digit numbers can be made with odd digits so that no two consecutive digits are same.
Q.23 It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many
such arrangements are possible?
Q.24 If no two books are alike, in how many ways can 2 red, 3 green, and 4 blue books be arranged on a
shelf so that all the books of the same colour are together?
Q.25 How many natural numbers are there with the property that they can be expressed as the sum of the
cubes of two natural numbers in two different ways.
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-2
Q.1
How many of the 900 three digit numbers have at least one even digit?
(A) 775
(B) 875
(C) 450
(D) 750
Q.2
The number of natual numbers from 1000 to 9999 (both inclusive) that do not have all 4 different digits is
(A) 4048
(B) 4464
(C) 4518
(D) 4536
OR
What can you say about the number of even numbers under the same constraints?
Q.3
The number of different seven digit numbers that can be written using only three digits
1, 2 & 3 under the condition that the digit 2 occurs exactly twice in each number is :
(A) 672
(B) 640
(C) 512
(D) none
Q.4
Out of seven consonants and four vowels, the number of words of six letters, formed by taking four
consonants and two vowels is (Assume that each ordered group of letter is a word):
(A) 210
(B) 462
(C) 151200
(D) 332640
Q.5
All possible three digits even numbers which can be formed with the condition that if 5 is one of the digit,
then 7 is the next digit is :
(A) 5
(B) 325
(C) 345
(D) 365
Q.6
For some natural N , the number of positive integral ' x ' satisfying the equation ,
1 ! + 2 ! + 3 ! + ...... + (x !) = (N)2 is :
(A) none
(B) one
(C) two
(D) infinite
Q.7
The number of six digit numbers that can be formed from the digits 1, 2, 3, 4, 5, 6 & 7 so that digits do
not repeat and the terminal digits are even is :
(A) 144
(B) 72
(C) 288
(D) 720
Q.8
A new flag is to be designed with six vertical strips using some or all of the colours yellow, green, blue
and red. Then, the number of ways this can be done such that no two adjacent strips have the same
colour is
(A) 12 × 81
(B) 16 × 192
(C) 20 × 125
(D) 24 × 216
Q.9
In how many ways can 5 colours be selected out of 8 different colours including red, blue, and green
(a)
if blue and green are always to be included,
(b)
if red is always excluded,
(c)
if red and blue are always included but green excluded?
Q.10 A 5 digit number divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 & 5 without repetition. The
total number of ways this can be done is :
(A) 3125
(B) 600
(C) 240
(D) 216
Q.11
Number of 9 digits numbers divisible by nine using the digits from 0 to 9 if each digit is used atmost once
is K . 8 ! , then K has the value equal to ______ .
Q.12 Number of natural numbers less than 1000 and divisible by 5 can be formed with the ten digits, each digit
not occuring more than once in each number is ______ .
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-3
Q.1
Find the number of ways in which letters of the word VALEDICTORY be arranged so that the vowels
may never be separated.
Q.2
How many numbers between 400 and 1000 (both exclusive) can be made with the digits 2,3,4,5,6,0 if
(a)
repetition of digits not allowed.
(b)
repetition of digits is allowed.
Q.3
Number of odd integers between 1000 and 8000 which have none of their digits repeated, is
(A) 1014
(B) 810
(C) 690
(D) 1736
Q.4
If 20Pr = 13× 20Pr–1 , then the value of r is ___________.
Q.5
The number of ways in which 5 different books can be distributed among 10 people if each person can
get at most one book is :
(A) 252
(B) 105
(C) 510
(D) 10C5.5!
Q.6
The product of all odd positive integers less than 10000, is
(A)
(10000)!
(5000!) 2
(B)
(10000)!
25000
(C)
(9999)!
25000
(D)
(10000)!
2
·(5000)!
5000
Q.7
The 9 horizontal and 9 vertical lines on an 8 × 8 chessboard form 'r' rectangles and 's' squares. The ratio
s
in its lowest terms is
r
1
17
4
(A)
(B)
(C)
(D) none
6
108
27
Q.8
There are 720 permutations of the digits 1, 2, 3, 4, 5, 6. Suppose these permutations are arranged from
smallest to largest numerical values, beginning from 1 2 3 4 5 6 and ending with 6 5 4 3 2 1.
(a) What number falls on the 124th position? (b) What is the position of the number 321546?
Q.9
A student has to answer 10 out of 13 questions in an examination . The number of ways in which he can
answer if he must answer atleast 3 of the first five questions is :
(A) 276
(B) 267
(C) 80
(D) 1200
Q.10 The number of three digit numbers having only two consecutive digits identical is
(A) 153
(B) 162
(C) 180
(D) 161
Q.11
Number of 3 digit numbers in which the digit at hundreath's place is greater than the other two digit is
(A) 285
(B) 281
(C) 240
(D) 204
Q.12 Number of permutations of 1, 2, 3, 4, 5, 6, 7, 8 and 9 taken all at a time are such that the digit
1 appearing somewhere to the left of 2
3 appearing to the left of 4 and
5 somewhere to the left of 6, is
(e.g. 815723946 would be one such permutation)
(A) 9 · 7!
(B) 8!
(C) 5! · 4!
(D) 8! · 4!
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-4
Q.1
A telegraph has x arms & each arm is capable of (x 1) distinct positions, including the position of rest.
The total number of signals that can be made is ______ .
Q.2
The interior angles of a regular polygon measure 150º each . The number of diagonals of the polygon is
(A) 35
(B) 44
(C) 54
(D) 78
Q.3
Number of different natural numbers which are smaller than two hundred million & using only the digits
1 or 2 is :
(A) (3) . 28 2
(B) (3) . 28 1
(C) 2 (29 1)
(D) none
Q.4
5 Indian & 5 American couples meet at a party & shake hands . If no wife shakes hands with her own
husband & no Indian wife shakes hands with a male, then the number of hand shakes that takes place in
the party is :
(A) 95
(B) 110
(C) 135
(D) 150
Q.5
The number of n digit numbers which consists of the digits 1 & 2 only if each digit is to be used atleast
once, is equal to 510 then n is equal to:
(A) 7
(B) 8
(C) 9
(D) 10
Q.6
Number of six digit numbers which have 3 digits even & 3 digits odd, if each digit is to be used atmost
once is ______ .
Q.7
The tamer of wild animals has to bring one by one 5 lions & 4 tigers to the circus arena. The number of
ways this can be done if no two tigers immediately follow each other is ______.
Q.8
18 points are indicated on the perimeter of a triangle ABC (see figure).
How many triangles are there with vertices at these points?
(A) 331
(B) 408
(C) 710
(D) 711
Q.9
An English school and a Vernacular school are both under one superintendent . Suppose that the
superintendentship, the four teachership of English and Vernacular school each, are vacant, if there be
altogether 11 candidates for the appointments, 3 of whom apply exclusively for the superintendentship
and 2 exclusively for the appointment in the English school, the number of ways in which the different
appointments can be disposed of is :
(A) 4320
(B) 268
(C) 1080
(D) 25920
Q.10 A committee of 5 is to be chosen from a group of 9 people. Number of ways in which it can be formed
if two particular persons either serve together or not at all and two other particular persons refuse to
serve with each other, is
(A) 41
(B) 36
(C) 47
(D) 76
Q.11
A question paper on mathematics consists of twelve questions divided into three parts A, B and C, each
containing four questions . In how many ways can an examinee answer five questions, selecting atleast
one from each part .
(A) 624
(B) 208
(C) 2304
(D) none
Q.12 If m denotes the number of 5 digit numbers if each successive digits are in their descending order of
magnitude and n is the corresponding figure, when the digits are in their ascending order of magnitude
then (m – n) has the value
(A) 10C4
(B) 9C5
(C) 10C3
(D) 9C3
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-5
Q.1
There are m points on a straight line AB & n points on the line AC none of them being the point A. Triangles
are formed with these points as vertices, when
(i) A is excluded
(ii) A is included. The ratio of number of triangles in the two cases is:
(A)
m n 2
m n
(B)
m n 2
m n 1
(C)
m n 2
m n 2
(D)
m (n 1)
(m 1) (n 1)
Q.2
Number of ways in which 7 green bottles and 8 blue bottles can be arranged in a row if exactly 1 pair of
green bottles is side by side, is (Assume all bottles to be alike except for the colour).
(A) 84
(B) 360
(C) 504
(D) 84
Q.3
In a certain algebraical exercise book there are 4 examples on arithmetical progressions, 5 examples on
permutation combination and 6 examples on binomial theorem . Number of ways a teacher can select
for his pupils atleast one but not more than 2 examples from each of these sets, is ______ .
Q.4
The kindergarten teacher has 25 kids in her class . She takes 5 of them at a time, to zoological garden as
often as she can, without taking the same 5 kids more than once. Find the number of visits, the teacher
makes to the garden and also the number of of visits every kid makes.
Q.5
There are n persons and m monkeys (m > n). Number of ways in which each person may become the
owner of one monkey is
(A) nm
(B) mn
(C) mPn
(D) mn
Q.6
Seven different coins are to be divided amongst three persons . If no two of the persons receive the same
number of coins but each receives atleast one coin & none is left over, then the number of ways in which
the division may be made is :
(A) 420
(B) 630
(C) 710
(D) none
Q.7
Let there be 9 fixed points on the circumference of a circle . Each of these points is joined to every one
of the remaining 8 points by a straight line and the points are so positioned on the circumference that
atmost 2 straight lines meet in any interior point of the circle . The number of such interior intersection
points is :
(A) 126
(B) 351
(C) 756
(D) none of these
Q.8
The number of 5 digit numbers such that the sum of their digits is even is :
(A) 50000
(B) 45000
(C) 60000
(D) none
Q.9
A forecast is to be made of the results of five cricket matches, each of which can be win, a draw or a loss
for Indian team. Find
the number of different possible forecasts
the number of forecasts containing 0, 1, 2, 3, 4 and 5 errors respectively
(i)
(ii)
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
Q.10 The number of ways in which 8 distinguishable apples can be distributed among 3 boys such that every
boy should get atleast 1 apple & atmost 4 apples is K · 7P3 where K has the value equal to
(A) 14
(B) 66
(C) 44
(D) 22
Q.11
A women has 11 close friends. Find the number of ways in which she can invite 5 of them to dinner, if two
particular of them are not on speaking terms & will not attend together.
Q.12 A rack has 5 different pairs of shoes. The number of ways in which 4 shoes can be chosen from it so that
there will be no complete pair is
(A) 1920
(B) 200
(C) 110
(D) 80
Paragraph for question nos. 13 to 15
Consider the word W = MISSISSIPPI
Q.13 If N denotes the number of different selections of 5 letters from the word W = MISSISSIPPI then N
belongs to the set
(A) {15, 16, 17, 18, 19}
(B) {20, 21, 22, 23, 24}
(C) {25, 26, 27, 28, 29}
(D) {30, 31, 32, 33, 34}
Q.14 Number of ways in which the letters of the word W can be arranged if atleast one vowel is separated
from rest of the vowels
8!·161
(A) 4!·4!·2!
8!·161
(B) 4 ·4!·2!
8!·161
(C) 4!·2!
8! 165
(D) 4!·2! · 4!
10!
Q.15 If the number of arrangements of the letters of the word W if all the S's and P's are separated is (K)
4!·4!
then K equals
(A)
6
5
(B) 1
(C)
4
3
(D)
3
2
Q.16 In how many different ways a grandfather along with two of his grandsons and four grand daughters can
be seated in a line for a photograph so that he is always in the middle and the two grandsons are never
adjacent to each other.
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-6
Q.1
Number of different ways in which 8 different books can be distributed among 3 students, if each student
receives atleast 2 books is ______.
Q.2
There are 10 seats in a double decker bus, 6 in the lower deck and 4 on the upper deck. Ten passengers
board the bus, of them 3 refuse to go to the upper deck and 2 insist on going up. The number of ways
in which the passengers can be accommodated is _____. (Assume all seats to be duly numbered)
Q.3
Find the number of permutations of the word "AUROBIND" in which vowels appear in an alphabetical
order.
Q.4
The greatest possible number of points of intersection of 9 different straight lines & 9 different circles in
a plane is:
(A) 117
(B) 153
(C) 270
(D) none
Q.5
An old man while dialing a 7 digit telephone number remembers that the first four digits consists of one
1's, one 2's and two 3's. He also remembers that the fifth digit is either a 4 or 5 while has no memorising
of the sixth digit, he remembers that the seventh digit is 9 minus the sixth digit. Maximum number of
distinct trials he has to try to make sure that he dials the correct telephone number, is
(A) 360
(B) 240
(C) 216
(D) none
Q.6
If as many more words as possible be formed out of the letters of the word "DOGMATIC" then the
number of words in which the relative order of vowels and consonants remain unchanged is ______ .
Q.7
Number of ways in which 7 people can occupy six seats, 3 seats on each side in a first class railway
compartment if two specified persons are to be always included and occupy adjacent seats on the same
side, is (5 !) · k then k has the value equal to :
(A) 2
(B) 4
(C) 8
(D) none
Q.8
Number of ways in which 9 different toys be distributed among 4 children belonging to different age
groups in such a way that distribution among the 3 elder children is even and the youngest one is to
receive one toy more, is :
9!
9!
(5!) 2
(A)
(B)
(C)
(D) none
2
8
3!(2!)3
Q.9
In an election three districts are to be canvassed by 2, 3 & 5 men respectively . If 10 men volunteer, the
number of ways they can be alloted to the different districts is :
(A)
10 !
2! 3! 5!
(B)
10 !
2! 5!
(C)
10 !
(2 !)2 5 !
(D)
10 !
(2 !)2 3 ! 5 !
Q.10 Let Pn denotes the number of ways in which three people can be selected out of ' n ' people sitting in
a row, if no two of them are consecutive. If , Pn + 1 Pn = 15 then the value of ' n ' is :
(A) 7
(B) 8
(C) 9
(D) 10
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Q.11
Permutation & Combination
Number of ways in which 8 people can be arranged in a line if A and B must be next each other and C
must be somewhere behind D, is equal to
(A) 10080
(B) 5040
(C) 5050
(D) 10100
Q.12 A has 3 maps and B has 9 maps. All the 12 maps being distinct. Determine the number of ways in which
they can exchange their maps if each keeps his initial number of maps.
Q.13 Number of three digit number with atleast one 3 and at least one 2 is
(A) 58
(B) 56
(C) 54
(D) 52
Paragraph for Question Nos. 14 to 16
16 players P1, P2, P3,.......P16 take part in a tennis tournament. Lower suffix player is better than any
higher suffix player. These players are to be divided into 4 groups each comprising of 4 players and the
best from each group is selected for semifinals.
Q.14 Number of ways in which 16 players can be divided into four equal groups, is
35 8
(2r 1)
(A)
27 r 1
35 8
(2r 1)
(B)
24 r 1
35 8
(2r 1)
(C)
52 r 1
35 8
(2r 1)
(D)
6 r 1
Q.15 Number of ways in which they can be divided into 4 equal groups if the players P1, P2, P3 and P4 are in
different groups, is :
(A)
(11)!
36
(B)
(11)!
72
(C)
(11)!
108
(D)
(11)!
216
Q.16 Number of ways in which these 16 players can be divided into four equal groups, such that when the
best player is selected from each group, P6 is one among them, is (k)
(A) 36
(B) 24
(C) 18
12!
. The value of k is :
( 4!) 3
(D) 20
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-7
Q.1
There are 10 red balls of different shades & 9 green balls of identical shades. Then the number of
arranging them in a row so that no two green balls are together is
(A) (10 !) . 11P9
(B) (10 !) . 11C9
(C) 10 !
(D) 10 ! 9 !
Q.2
Number of ways in which n distinct objects can be kept into two identical boxes so that no box remains
empty, is ______ .
Q.3
A shelf contains 20 different books of which 4 are in single volume and the others form sets of 8, 5 and
3 volumes respectively. Number of ways in which the books may be arranged on the shelf, if the
volumes of each set are together and in their due order is
20!
(A)
(B) 7!
(C) 8!
(D) 7 . 8!
8! 5! 3!
Q.4
If all the letters of the word "QUEUE" are arranged in all possible manner as they are in a dictionary,
then the rank of the word QUEUE is :
(A) 15th
(B) 16th
(C) 17th
(D) 18th
Q.5
Number of rectangles in the grid shown which are not squares is
(A) 160
(B) 162
(C) 170
(D) 185
Q.6
All the five digit numbers in which each successive digit exceeds its predecessor are arranged in the
increasing order of their magnitude. The 97th number in the list does not contain the digit
(A) 4
(B) 5
(C) 7
(D) 8
Q.7
The number of combination of 16 things, 8 of which are alike and the rest different, taken 8 at a time is _____.
Q.8
The number of different ways in which five 'dashes' and eight 'dots' can be arranged, using only seven of
these 13 'dashes' & 'dots' is :
(A) 1287
(B) 119
(C) 120
(D) 1235520
Q.9
In a certain college at the B.Sc. examination, 3 candidates obtained first class honours in each of the
following subjects: Physics, Chemistry and Maths, no candidates obtaining honours in more than one subject;
Number of ways in which 9 scholarships of different value be awarded to the 9 candidates if due regard is
to be paid only to the places obtained by candidates in any one subject is __________.
Q.10 There are n identical red balls & m identical green balls . The number of different linear arrangements
consisting of "n red balls but not necessarily all the green balls" is xCy then
(A) x = m + n, y = m (B) x = m + n + 1, y = m (C) x = m + n + 1, y = m + 1 (D) x = m + n , y = n
Direction for Q.11 & Q.12
In how many ways the letters of the word “COMBINATORICS” can be arranged if
Q.11
All the vowels are always grouped together to form a contiguous block.
Q.12 All vowels and all consonants are alphabetically ordered.
Q.13 How many different arrangements are possible with the factor of the term a2b4c5 written at full length.
Q.14 Find the number of 4 digit numbers starting with 1 and having exactly two identical digits.
Q.15 Number of ways in which 5 A's and 6 B's can be arranged in a row which reads the same backwards
and forwards, is
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-8
Q.1
Number of different words that can be formed using all the letters of the word "DEEPMALA" if two
vowels are together and the other two are also together but separated from the first two is :
(A) 960
(B) 1200
(C) 2160
(D) 1440
Q.2
The number of ways in which 10 boys can take positions about a round table if two particular boys must
not be seated side by side is :
(A) 10 (9) !
(B) 9 (8) !
(C) 7 (8) !
(D) none
Q.3
In a unique hockey series between India & Pakistan, they decide to play on till a team wins 5 matches .
The number of ways in which the series can be won by India, if no match ends in a draw is :
(A) 126
(B) 252
(C) 225
(D) none
Q.4
Number of cyphers at the end of
(A) 0
(B) 1
2002C
1001
is
(C) 2
(D) 200
Q.5
Three vertices of a convex n sided polygon are selected. If the number of triangles that can be constructed
such that none of the sides of the triangle is also the side of the polygon is 30, then the polygon is a
(A) Heptagon
(B) Octagon
(C) Nonagon
(D) Decagon
Q.6
A gentleman invites a party of m + n (m n) friends to a dinner & places m at one table T1 and n at
another table T2 , the table being round . If not all people shall have the same neighbour in any two
arrangement, then the number of ways in which he can arrange the guests, is
(m n) !
1 ( m n) !
( m n) !
(A)
(B)
(C) 2
(D) none
4 mn
mn
2
mn
Q.7
There are 12 guests at a dinner party . Supposing that the master and mistress of the house have fixed
seats opposite one another, and that there are two specified guests who must always, be placed next to
one another ; the number of ways in which the company can be placed, is:
(A) 20 . 10 !
(B) 22 . 10 !
(C) 44 . 10 !
(D) none
Q.8
Let Pn denotes the number of ways of selecting 3 people out of ' n ' sitting in a row , if no two of them
are consecutive and Qn is the corresponding figure when they are in a circle . If Pn Qn = 6 , then
' n ' is equal to :
(A) 8
(B) 9
(C) 10
(D) 12
Q.9
Define a 'good word' as a sequence of letters that consists only of the letters A, B and C and in which A
never immidiately followed by B, B is never immediately followed by C, and C is never immediately
followed by A. If the number of n-letter good words are 384, find the value of n.
Q.10 Six married couple are sitting in a room. Find the number of ways in which 4 people can be selected so that
(a)
they do not form a couple
(b)
they form exactly one couple
(c)
they form at least one couple
(d)
they form atmost one couple
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Q.11
Permutation & Combination
Fifty college teachers are surveyed as to their possession of colour TV, VCR and tape recorder. Of
them, 22 own colour TV, 15 own VCR and 14 own tape recorders. Nine of these college teachers own
exactly two items out of colour TV, VCR and tape recorders ; and, one college teacher owns all three.
how many of the 50 college teachers own none of three, colour TV, VCR or tape recorder?
(A) 4
(B) 9
(C) 10
(D) 11
Q.12 There are counters available in x different colours. The counters are all alike except for the colour. The
total number of arrangements consisting of y counters, assuming sufficient number of counters of each
colour, if no arrangement consists of all counters of the same colour is :
(A) xy x
(B) xy y
(C) yx x
(D) yx y
Q.13 There are (p + q) different books on different topics in Mathematics. (p q)
If L = The number of ways in which these books are distributed between two students X and Y such
that X get p books and Y gets q books.
M = The number of ways in which these books are distributed between two students X and Y such that
one of them gets p books and another gets q books.
N = The number of ways in which these books are divided into two groups of p books and q books then,
(A) L = M = N
(B) L = 2M = 2N
(C) 2L = M = 2N
(D) L = M = 2N
The question given below contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason).
Question has 4 choices (A), (B), (C) and (D), out of which ONLY ONE is correct. Choose the
correct alternative.
Q.14 Statement 1: The sum 40C0 · 60C10 + 40C1 · 60C9 + ......... + 40C10 · 60C0 equals 100C10.
because
Statement 2: Number of ways of selecting 10 students out of 40 boys and 60 girls is 100C10.
(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true.
MATCH THE COLUMN
Q.15
(A)
Column-I
In a plane a set of 8 parallel lines intersect a set of n parallel lines,
that goes in another direction, forming a total of 1260 parallelograms.
The value of n is equal to
n 1
P3
If
(C)
Number of ways in which 5 persons A, B, C, D and E
can be seated on round table if A and D do not sit next
to each other
Number of cyphers at the end of the number 50P25
(D)
n
P4
=
1
then n is equal to
9
(B)
Column-II
(P)
6
(Q)
9
(R)
10
(S)
12
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-9
n
n 1
Q.1
nC
r 0
r
Cr
n C r1
n ( n 1)
(A) 2 (n 1)
is equal to
(B)
n 1
2
(C)
n (n 1)
2
(D)
n
2
Q.2
Let m denote the number of ways in which 4 different books are distributed among 10 persons, each
receiving none or one only and let n denote the number of ways of distribution if the books are all alike.
Then :
(A) m = 4n
(B) n = 4m
(C) m = 24n
(D) none
Q.3
The number of ways in which we can arrange n ladies & n gentlemen at a round table so that 2 ladies or
2 gentlemen may not sit next to one another is :
(A) (n 1) ! (n 2) !
(B) (n !) (n 1) !
(C) (n + 1) ! (n) !
(D) none
Q.4
There are six periods in each working day of a school. Number of ways in which 5 subjects can be
arranged if each subject is allotted at least one period and no period remains vacant is
(A) 210
(B) 1800
(C) 360
(D) 120
Q.5
The number of all possible selections of one or more questions from 10 given questions, each equestion
having an alternative is :
(A) 310
(B) 210 1
(C) 310 1
(D) 210
Q.6
A team of 8 students goes on an excursion, in two cars, of which one can seat 5 and the other only 4. If
internal arrangement inside the car does not matter then the number of ways in which they can travel, is
(A) 91
(B) 182
(C) 126
(D) 3920
Q.7
The number of divisors of the number 21600 is _____ and the sum of these divisors is ______.
Q.8
10 IIT & 2 PET students sit in a row. The number of ways in which exactly 3 IIT students sit between 2
PET student is ______ .
Q.9
The number of ways of choosing a committee of 2 women & 3 men from 5 women & 6 men, if Mr. A
refuses to serve on the committee if Mr. B is a member & Mr. B can only serve, if Miss C is the member
of the committee, is :
(A) 60
(B) 84
(C) 124
(D) none
Q.10 Six persons A, B, C, D, E and F are to be seated at a circular table . The number of ways this can be
done if A must have either B or C on his right and B must have either C or D on his right is :
(A) 36
(B) 12
(C) 24
(D) 18
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Q.11
Permutation & Combination
There are 2 identical white balls, 3 identical red balls and 4 green balls of different shades. The number
of ways in which they can be arranged in a row so that atleast one ball is separated from the balls of the
same colour, is :
(A) 6 (7 ! 4 !)
(B) 7 (6 ! 4 !)
(C) 8 ! 5 !
(D) none
Q.12 Sameer has to make a telephone call to his friend Harish, Unfortunately he does not remember the 7 digit
phone number. But he remembers that the first three digits are 635 or 674, the number is odd and there
is exactly one 9 in the number. The maximum number of trials that Sameer has to make to be successful
is
(A) 10,000
(B) 3402
(C) 3200
(D) 5000
Q.13 Six people are going to sit in a row on a bench. A and B are adjacent, C does not want to sit adjacent
to D. E and F can sit anywhere. Number of ways in which these six people can be seated, is
(A) 200
(B) 144
(C) 120
(D) 56
Q.14 Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle . Then the
maximum number of circles that can be drawn so that each contains atleast three of the given points is :
(A) 216
(B) 156
(C) 172
(D) none
Q.15 One hundred management students who read at least one of the three business magazines are surveyed
to study the readership pattern. It is found that 80 read Business India, 50 read Business world, and 30
read Business Today. Five students read all the three magazines. How many read exactly two magazines?
(A) 50
(B) 10
(C) 95
(D) 25
Q.16 Find the number of 10 digit numbers using the digits 0, 1, 2, ....... 9 without repetition. How many of
these are divisible by 4.
Q.17 A four digit number is called a doublet if any of its digit is the same as only one neighbour. For example,
1221 is a doublet but 1222 is not. Number of such doublets are
(A) 2259
(B) 2268
(C) 2277
(D) 2349
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP NO.-10
Choose the correct alternative (only one is correct):
Q.1
There are 100 different books in a shelf. Number of ways in which 3 books can be selected so that no
two of which are neighbours is
(A) 100C3 – 98
(B) 97C3
(C) 96C3
(D) 98C3
Q.2
Two classrooms A and B having capacity of 25 and (n–25) seats respectively.An denotes the number of
possible seating arrangements of room 'A', when 'n' students are to be seated in these rooms, starting
from room 'A' which is to be filled up full to its capacity.
If An – An–1 = 25! (49C25) then 'n' equals
(A) 50
(B) 48
(C) 49
(D) 51
Q.3
The sum of all numbers greater than 1000 formed by using digits 1, 3, 5, 7 no digit being repeated in any
number is :
(A) 72215
(B) 83911
(C) 106656
(D) 114712
Q.4
Number of positive integral solutions satisfying the equation (x1 + x2 + x3) (y1 + y2) = 77, is
(A) 150
(B) 270
(C) 420
(D) 1024
Q.5
Distinct 3 digit numbers are formed using only the digits 1, 2, 3 and 4 with each digit used at most once
in each number thus formed. The sum of all possible numbers so formed is
(A) 6660
(B) 3330
(C) 2220
(D) none
Q.6
The streets of a city are arranged like the lines of a chess board . There are m streets running North to
South & 'n' streets running East to West . The number of ways in which a man can travel from NW to SE
corner going the shortest possible distance is :
(A) m2 n 2
(B) (m 1)2 . (n 1)2 (C)
( m n) !
m! . n!
(D)
( m n 2) !
( m 1) ! . ( n 1) !
Q.7
An ice cream parlour has ice creams in eight different varieties . Number of ways of choosing 3 ice
creams taking atleast two ice creams of the same variety, is :
(A) 56
(B) 64
(C) 100
(D) none
(Assume that ice creams of the same variety are identical & available in unlimited supply)
Q.8
There are 12 books on Algebra and Calculus in our library , the books of the same subject being
different. If the number of selections each of which consists of 3 books on each topic is greatest then the
number of books of Algebra and Calculus in the library are respectively:
(A) 3 and 9
(B) 4 and 8
(C) 5 and 7
(D) 6 and 6
Q.9
The sum of all the numbers formed from the digits 1, 3, 5, 7, 9 which are smaller than 10,000 if repetion
of digits is not allowed, is
(A) (28011)S
(B) (28041)S
(C) (28121)S
(D) (29152)S
where S = (1+3+5+7+9)
Choose the correct alternatives (More than one are correct):
Q.10 The combinatorial coefficient C(n, r) is equal to
(A) number of possible subsets of r members from a set of n distinct members.
(B) number of possible binary messages of length n with exactly r 1's.
(C) number of non decreasing 2-D paths from the lattice point (0, 0) to (r, n).
(D) number of ways of selecting r things out of n different things when a particular thing is always
included plus the number of ways of selecting 'r' things out of n, when a particular thing is always
excluded.
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Q.11
Permutation & Combination
Identify the correct statement(s).
(A) Number of naughts standing at the end of
is 30 .
(B) A telegraph has 10 arms and each arm is capable of 9 distinct positions excluding the position of rest.
The number of signals that can be transmitted is 1010 1 .
(C) Number of numbers greater than 4 lacs which can be formed by using only the digits 0, 2, 2, 4,
4 and 5 is 90.
(D) In a table tennis tournament, every player plays with every other player. If the number of games
played is 5050 then the number of players in the tournament is 100.
Q.12 There are 10 questions, each question is either True or False. Number of different sequences of incorrect
answers is also equal to
(A) Number of ways in which a normal coin tossed 10 times would fall in a definite order if both Heads
and Tails are present.
(B) Number of ways in which a multiple choice question containing 10 alternatives with one or more than
one correct alternatives, can be answered.
(C) Number of ways in which it is possible to draw a sum of money with 10 coins of different denominations
taken some or all at a time.
(D) Number of different selections of 10 indistinguishable things taken some or all at a time.
Q.13 The continued product, 2 . 6 . 10 . 14 ...... to n factors is equal to
(A) 2nCn
(B) 2nPn
(C) (n + 1) (n + 2) (n + 3) ...... (n + n)
(D) none
Q.14 The Number of ways in which five different books to be distributed among 3 persons so that each
person gets at least one book, is equal to the number of ways in which
(A) 5 persons are allotted 3 different residential flats so that and each person is alloted at most one flat
and no two persons are alloted the same flat.
(B) number of parallelograms (some of which may be overlapping) formed by one set of 6 parallel lines
and other set of 5 parallel lines that goes in other direction.
(C) 5 different toys are to be distributed among 3 children, so that each child gets at least one toy.
(D) 3 mathematics professors are assigned five different lecturers to be delivered , so that each professor
gets at least one lecturer.
Q.15 The combinatorial coefficient n – 1Cp denotes
(A) the number of ways in which n things of which p are alike and rest different can be arranged in a circle.
(B) the number of ways in which p different things can be selected out of n different thing if a particular
thing is always excluded.
(C) number of ways in which n alike balls can be distributed in p different boxes so that no box remains
empty and each box can hold any number of balls.
(D) the number of ways in which (n – 2) white balls and p black balls can be arranged in a line if black
balls are separated, balls are all alike except for the colour.
Q.16 The maximum number of permutations of 2n letters in which there are only a's & b's, taken all at a time
is given by :
(A)
2nC
(C)
n 1 n 2 n 3 n 4
2n 1 2n
.
.
.
......
.
1
2
3
4
n1 n
n
(B)
(D)
2 6 10
4n 6 4n 2
. .
......
.
1 2 3
n 1
n
2 n . 1 . 3 . 5 ...... (2 n 3) (2 n 1)
n!
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
Q.17 Number of ways in which 3 numbers in A.P. can be selected from 1, 2, 3, ...... n is :
n 1
2
(A)
(C)
n 12
4
2
if n is even
if n is odd
(B)
n n 2
if n is odd
4
n n 2
(D)
if n is even
4
Q.18 If P(n, n) denotes the number of permutations of n different things taken all at a time then P(n, n) is also
identical to
(A) r! · P(n, n – r)
(B) (n – r) · P(n, r)
(C) n · P(n – 1, n – 1) (D) P(n, n – 1)
where 0 r n
Q.19 Which of the following statements are correct?
(A) Number of words that can be formed with 6 only of the letters of the word "CENTRIFUGAL" if
each word must contain all the vowels is 3 · 7!
(B) There are 15 balls of which some are white and the rest black. If the number of ways in which the
balls can be arranged in a row, is maximum then the number of white balls must be equal to 7 or 8.
Assume balls of the same colour to be alike.
(C) There are 12 things, 4 alike of one kind, 5 alike and of another kind and the rest are all different. The
total number of combinations is 240.
(D) Number of selections that can be made of 6 letters from the word "COMMITTEE" is 35.
MATCH THE COLUMN
Q.20
(A)
(B)
(C)
(D)
Q.21
(A)
(B)
(C)
(D)
(E)
Column I
Number of increasing permutations of m symbols are there from the n set
numbers {a1, a2, , an} where the order among the numbers is given by
a1 < a2 < a3 < an–1 < an is
There are m men and n monkeys. Number of ways in which every monkey
has a master, if a man can have any number of monkeys
Number of ways in which n red balls and (m – 1) green balls can be arranged
in a line, so that no two red balls are together, is
(balls of the same colour are alike)
Number of ways in which 'm' different toys can be distributed in 'n' children
if every child may receive any number of toys, is
Column II
(P)
nm
Column-I
Four different movies are running in a town. Ten students go to watch
these four movies. The number of ways in which every movie is watched
by atleast one student, is
(Assume each way differs only by number of students watching a movie)
Consider 8 vertices of a regular octagon and its centre. If T denotes the
number of triangles and S denotes the number of straight lines that can
be formed with these 9 points then the value of (T – S) equals
In an examination, 5 children were found to have their mobiles in their
pocket. The Invigilator fired them and took their mobiles in his possession.
Towards the end of the test, Invigilator randomly returned their mobiles. The
number of ways in which at most two children did not get their own mobiles is
The product of the digits of 3214 is 24. The number of 4 digit natural
numbers such that the product of their digits is 12, is
The number of ways in which a mixed double tennis game can be
arranged from amongst 5 married couple if no husband & wife plays
in the same game, is
Column-II
(P)
11
(Q)
mC
(R)
nC
(S)
mn
(Q)
36
(R)
52
(S)
60
(T)
84
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n
m
27
Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
SUBJECTIVE :
Q.22 A commitee of 10 members is to be formed with members chosen from the faculties of Arts, Economics,
Education, Engineering, Medicine and Science. Number of possible ways in which the faculties
representation be distributed on this committee, is ________.
(Assume every department contains more than 10 members).
Q.23 How many ways are there to seat n married couples (n 3) around a table such that men and women
alternate and each women is not adjacent to her husband.
Q.24 10 identical ball are distributed in 5 different boxes kept in a row and labled A, B, C, D and E. Find the
number of ways in which the ball can be distributed in the boxes if no two adjacent boxes remain empty.
Q.25 The number of non negative integral solution of the inequation x + y + z + w 7 is ____ .
Q.26 On the normal chess board as shown, I1 & I2 are two insects which starts moving towards each
other. Each insect moving with the same constant speed. Insect I1 can move only to the right or
upward along the lines while the insect I2 can move only to the left or downward along the lines of the
chess board. Prove that the total number of ways the two insects can meet at same point during their
trip is equal to
9 10 11 12 13 14 15 16
8 7 6 5 4 3 2 1
OR
1 3 5 7 9 11 13 15
1 2 3 4 5 6 7 8
28
OR
2 6 10 14 18 22 26 30
1 2 3 4 5 6 7 8
Q.27 How many numbers gretater than 1000 can be formed from the digits 112340 taken 4 at a time.
Q.28 Tom has 15 ping-pong balls each uniquely numbered from 1 to 15. He also has a red box, a blue box,
and a green box.
(a) How many ways can Tom place the 15 distinct balls into the three boxes so that no box is empty?
(b) Suppose now that Tom has placed 5 ping-pong balls in each box. How many ways can he choose
5 balls from the three boxes so that he chooses at least one from each box?
Q.29 Find the number of ways in which 12 identical coins can be distributed in 6 different purses, if not more
than 3 & not less than 1 coin goes in each purse.
Q.30 A drawer is fitted with n compartments and each compartment contains n counter, no two of which
marked alike. Number of combinations which can be made with these counters if no two out of the same
compartment enter into any combination, is ______ .
Q.31 In how many ways it is possible to select six letters, including at least one vowel from the letters of the
word "F L A B E L L I F O R M ". (It is a picnic spot in U.S.A.)
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
EXERCISE–I
The straight lines l1 , l2 & l3 are parallel & lie in the same plane. A total of m points are taken on the line
l1 , n points on l2 & k points on l3. How many maximum number of triangles are there whose vertices are
at these points ?
2. (a) How many five digits numbers divisible by 3 can be formed using the digits 0, 1, 2, 3, 4, 7 and 8 if each
digit is to be used atmost once.
(b) Find the number of 4 digit positive integers if the product of their digits is divisible by 3.
3.
There are 2 women participating in a chess tournament. Every participant played 2 games with the other
participants. The number of games that the men played between themselves exceeded by 66 as compared
to the number of games that the men played with the women. Find the number of participants & the total
numbers of games played in the tournament.
4.
All the 7 digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once, and not divisible by
5 are arranged in the increasing order. Find the (2004)th number in this list.
5.
5 boys & 4 girls sit in a straight line. Find the number of ways in which they can be seated if 2 girls are
together & the other 2 are also together but separate from the first 2.
6.
A crew of an eight oar boat has to be chosen out of 11 men five of whom can row on stroke side only, four
on the bow side only, and the remaining two on either side. How many different selections can be made ?
7.
An examination paper consists of 12 questions divided into parts A & B.
Part-A contains 7 questions & PartB contains 5 questions. A candidate is required to attempt 8 questions
selecting atleast 3 from each part. In how many maximum ways can the candidate select the questions ?
8.
In how many ways can a team of 6 horses be selected out of a stud of 16 , so that there shall always be
3 out of A B C A B C , but never A A , B B or C C together.
9.
During a draw of lottery, tickets bearing numbers 1, 2, 3,......, 40, 6 tickets are drawn out & then
arranged in the descending order of their numbers. In how many ways, it is possible to have 4th ticket
bearing number 25.
10.
Find the number of distinct natural numbers upto a maximum of 4 digits and divisible by 5, which can be
formed with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each digit not occuring more than once in each number.
11.
The Indian cricket team with eleven players, the team manager, the physiotherapist and two umpires are to
travel from the hotel where they are staying to the stadium where the test match is to be played. Four of them
residing in the same town own cars, each a four seater which they will drive themselves. The bus which was to
pick them up failed to arrive in time after leaving the opposite team at the stadium. In how many ways can they
be seated in the cars ? In how many ways can they travel by these cars so as to reach in time, if the seating
arrangement in each car is immaterial and all the cars reach the stadium by the same route.
12.
There are n straight lines in a plane, no 2 of which parallel , & no 3 pass through the same point. Their
point of intersection are joined. Show that the number of maximum fresh lines thus introduced is
n ( n 1)(n 2)( n 3)
.
8
13.
In how many ways can you divide a pack of 52 cards equally among 4 players. In how many ways the
cards can be divided in 4 sets, 3 of them having 17 cards each & the 4th with 1 card.
1.
14.
A firm of Chartered Accountants in Bombay has to send 10 clerks to 5 different companies, two clerks
in each. Two of the companies are in Bombay and the others are outside. Two of the clerks prefer to
work in Bombay while three others prefer to work outside. In how many ways can the assignment be
made if the preferences are to be satisfied.
15.
A train going from Cambridge to London stops at nine intermediate stations. 6 persons enter the train
during the journey with 6 different tickets of the same class. How many different sets of ticket may they
have had ?
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
16.
How many arrangements each consisting of 2 vowels & 2 consonants can be made out of the letters of
the word ‘DEVASTATION’ ?
17.
Find the number of ways in which the letters of the word 'KUTKUT' can be arranged so that no two
alike letters are together.
18.
Find the number of words each consisting of 3 consonants & 3 vowels that can be formed from the
letters of the word “Circumference”. In how many of these c’s will be together.
19.
There are 5 white , 4 yellow , 3 green , 2 blue & 1 red ball. The balls are all identical except for colour.
These are to be arranged in a line in 5 places. Find the number of distinct arrangements.
20.
How many 4 digit numbers are there which contains not more than 2 different digits ?
21.
An 8 oared boat is to be manned by a crew chosen from 14 men of which 4 can only steer but can not
row & the rest can row but cannot steer. Of those who can row, 2 can row on the bow side. In how
many ways can the crew be arranged.
22. (a) A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time, or any
combination of 1's and 2's. Find the total number of ways in which the person can go up the stairs.
(b) You walk up 12 steps, going up either 1 or 2 steps with each stride. There is a snake on the 8th step, so
you can not step there. Number of ways you can go up.
23.
Each of 3 committees has 1 vacancy which is to be filled from a group of 6 people. Find the number of
ways the 3 vacancies can be filled if ;
(i)
Each person can serve on atmost 1 committee.
(ii)
There is no restriction on the number of committees on which a person can serve.
(iii)
Each person can serve on atmost 2 committees.
24.
How many ten digit whole numbers satisfy the following property they have 2 and 5 as digits, and there
are no consecutive 2's in the number (i.e. any two 2's are separated by at least one 5).
25.
In how many other ways can the letters of the word MULTIPLE be arranged;
(i)
without changing the order of the vowels
(ii)
keeping the position of each vowel fixed &
(iii)
without changing the relative order/position of vowels & consonants.
26.
12 persons are to be seated at a square table, three on each side. 2 persons wish to sit on the north side
and two wish to sit on the east side. One other person insists on occupying the middle seat (which may
be on any side). Find the number of ways they can be seated.
27.
How many integers between 1000 and 9999 have exactly one pair of equal digit such as 4049 or 9902
but not 4449 or 4040 ?
28.
Determine the number of paths from the origin to the point (9, 9) in the cartesian plane which never pass
through (5, 5) in paths consisting only of steps going 1 unit North and 1 unit East.
29.
(i)
Prove that : nPr = n1Pr + r. n1Pr1
(ii)
If 20Cr+2 = 20C2r3 find 12Cr
(iii)
Prove that
(iv)
Find r if
30.
n1C
n1C
3+
4
15C = 15C
3r
r+3
> nC3 if n > 7.
There are 20 books on Algebra & Calculus in our library. Prove that the greatest number of selections
each of which consists of 5 books on each topic is possible only when there are 10 books on each topic
in the library.
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
EXERCISE–II
1.
2.
3.
4.
5.
6.
7.
How many 6 digits odd numbers greater than 60,0000 can be formed from the digits 5, 6, 7, 8, 9, 0 if
(i) repetitions are not allowed
(ii) repetitions are allowed.
A man has 3 friends. In how many ways he can invite one friend everyday for dinner on 6 successive
nights so that no friend is invited more than 3 times.
Find the number of 7 lettered words each consisting of 3 vowels and 4 consonants which can be formed
using the letters of the word "DIFFERENTIATION".
Find the number of ways in which 3 distinct numbers can be selected from the set
{31, 32, 33, ....... 3100, 3101} so that they form a G.P.
There are 2n guests at a dinner party. Supposing that the master and mistress of the house have fixed
seats opposite one another, and that there are two specified guests who must not be placed next to one
another. Show that the number of ways in which the company can be placed is (2n 2)!.(4n2 6n + 4).
The members of a chess club took part in a round robin competition in which each plays every one else
once. All members scored the same number of points, except four juniors whose total score were 17.5.
How many members were there in the club? Assume that for each win a player scores 1 point , for draw
1/2 point and zero for losing.
In an election for the managing committee of a reputed club , the number of candidates contesting
elections exceeds the number of members to be elected by r (r > 0). If a voter can vote in 967 different
ways to elect the managing committee by voting atleast 1 of them & can vote in 55 different ways to elect
(r 1) candidates by voting in the same manner. Find the number of candidates contesting the elections
& the number of candidates losing the elections.
Instruction for question nos. 8 to 10
2 Americal men; 2 British men; 2 Chinese men and one each of Dutch, Egyptial, French and German
persons are to be seated for a round table conference.
8.
If the number of ways in which they can be seated if exactly two pairs of persons of same nationality are
together is p(6!), then find p.
9.
If the number of ways in which only American pair is adjacent is equal to q(6!), then find q.
10.
If the number of ways in which no two people of the same nationality are together given by r (6!), find r.
11.
For each positive integer k, let Sk denote the increasing arithmetic sequence of integers whose first term
is 1 and whose common difference is k. For example, S3 is the sequence 1, 4, 7, 10...... Find the number
of values of k for which Sk contain the term 361.
12.
A shop sells 6 different flavours of ice-cream. In how many ways can a customer choose 4 ice-cream cones if
(i)
they are all of different flavours
(ii)
they are non necessarily of different flavours
(iii)
they contain only 3 different flavours (iv)
they contain only 2 or 3 different flavours ?
13.
There are n triangles of positive area that have one vertex A(0, 0) and the other two vertices whose
coordinates are drawn independently with replacement from the set {0, 1, 2, 3, 4} e.g. (1, 2),
(0, 1), (2, 2) etc. Find the value of n.
14. (a) How many divisors are there of the number x = 21600. Find also the sum of these divisors.
(b) In how many ways the number 7056 can be resolved as a product of 2 factors.
(c) Find the number of ways in which the number 300300 can be split into 2 factors which are relatively prime.
(d) Find the number of positive integers that are divisors of atleast one of the numbers 1010 ; 157 ; 1811.
15.
How many 15 letter arrangements of 5 A's, 5 B's and 5 C's have no A's in the first 5 letters, no B's in the
next 5 letters, and no C's in the last 5 letters.
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
How many different ways can 15 Candy bars be distributed between Ram, Shyam, Ghanshyam and
Balram, if Ram can not have more than 5 candy bars and Shyam must have at least two. Assume all
Candy bars to be alike.
17.
Find the number of three digits numbers from 100 to 999 inclusive which have any one digit that is the
average of the other two.
18.
Find the number of distinct throws which can be thrown with 'n' six faced normal dice which are
indistinguishable among themselves.
19.
There are 15 rowing clubs; two of the clubs have each 3 boats on the river; five others have each 2 and
the remaining eight have each 1; find the number of ways in which a list can be formed of the order of the
24 boats, observing that the second boat of a club cannot be above the first and the third above the
second. How many ways are there in which a boat of the club having single boat on the river is at the
third place in the list formed above?
20.
Consider a 7 digit telephone number 336 - 7624 which has the property that the first three digit prefix,
336 equals the product of the last four digits. How many seven digit phone numbers beginning with 336
have this property, e.g. (336–7624)
21.
Find the sum of all numbers greater than 10000 formed by using the digits 0 , 1 , 2 , 4 , 5 no digit being
repeated in any number.
22.
There are 3 cars of different make available to transport 3 girls and 5 boys on a field trip. Each car can
hold up to 3 children. Find
(a) the number of ways in which they can be accomodated.
(b) the numbers of ways in which they can be accomodated if 2 or 3 girls are assigned to one of the cars.
In both the cases internal arrangement of children inside the car is considered to be immaterial.
23.
Find the number of three elements sets of positive integers {a, b, c} such that a × b × c = 2310.
24.
Find the number of integer between 1 and 10000 with at least one 8 and at least one 9 as digits.
16.
EXERCISE–III
1.
2.
3.
4.
5.
6.
How many different nine digit numbers can be formed from the number 223355888 by rearranging its
digits so that the odd digits occupy even positions ?
[JEE '2000, (Scr)]
(A) 16
(B) 36
(C) 60
(D) 180
Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon of
' n ' sides. If Tn + 1 Tn = 21 , then ' n ' equals:
[JEE '2001, (Scr)]
(A) 5
(B) 7
(C) 6
(D) 4
The number of arrangements of the letters of the word BANANA in which the two N’s do not appear
adjacently is
[JEE 2002 (Screening), 3]
(A) 40
(B) 60
(C) 80
(D) 100
Number of points with integral co-ordinates that lie inside a triangle whose co-ordinates are
(0, 0), (0, 21) and (21,0)
[JEE 2003 (Screening), 3]
(A) 210
(B) 190
(C) 220
(D) None
Using permutation or otherwise, prove that
(n 2 ) !
( n!) n
is an integer, where n is a positive integer..
[JEE 2004, 2 out of 60]
A rectangle with sides 2m – 1 and 2n – 1 is divided into squares of unit length by
drawing parallel lines as shown in the diagram, then the number of rectangles
possible with odd side lengths is
(A) (m + n + 1)2
(B) 4m + n – 1
(C) m2n2
(D) mn(m + 1)(n + 1)
[JEE 2005 (Screening), 3]
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32
Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
7.
If r, s, t are prime numbers and p, q are the positive integers such that their LCM of p, q is is r2t4s2, then
the numbers of ordered pair of (p, q) is
(A) 252
(B) 254
(C) 225
(D) 224
[JEE 2006, 3]
8.
The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical
order as in an English dictionary. The number of words that appear before the word COCHIN is
(A) 360
(B) 192
(C) 96
(D) 48
[JEE 2007, 3]
Consider all possible permutations of the letters of the word ENDEANOEL
Match the statements / Expression in Column-I with the statements / Expressions in Column-II.
Column-I
Column-II
(A)
The number of permutations containing the word ENDEA is
(P)
5!
(B)
The number of permutations in which the letter E occurs in the
(Q)
2 × 5!
first and the last position is
(C)
The number of permutations in which none of the letters D, L, N
(R)
7 × 5!
occurs in the last five positions is
(D)
The number of permutations in which the letters A, E, O occurs
(S)
21 × 5!
only in odd positions is
[JEE 2008, 6]
9.
10.
The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits
1, 2 and 3 only, is
(A) 55
(B) 66
(C) 77
(D) 88
[JEE 2009, 3]
11.
Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to
(A) 25
(B) 34
(C) 42
(D) 41
[JEE 2010]
12.
The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that
each person gets at least one balls is
[JEE 2012]
(A) 75
(B) 150
(C) 210
(D) 243
13.
Let Tn be the number of all possible triangles formed by joining vertices of an n-shaded regular polygon.
If Tn+1 – Tn = 10, then the value of n is
[IIT JEE Main 2013]
(A) 8
(B) 7
(C) 5
(D) 10
14.
Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that
each envelope contains exactly one card and no card is placed in the envelope bearing the same number
and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways
it can be done is :
[IIT JEE Adv. 2014]
(A) 264
(B) 265
(C) 53
(D) 67
15.
Let n1 < n2 < n3 < n4 < n5 be positive integers such that n1 + n2 + n3 + n4 + n5 = 20. Then the number
of such distinct arrangements (n1, n2, n3, n4, n5) is
[IIT JEE Adv. 2014]
16.
The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without
repetition, is
[IIT JEE Main 2015]
(A) 72
(B) 216
(C) 192
(D) 120
17.
Let A and B be two sets containing four and two elements respectively. Then the number of subsets of
the set A × B, each having at least three elements is :
[IIT JEE Main 2015]
(A) 510
(B) 219
(C) 256
(D) 275
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
18.
Permutation & Combination
Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the
girls stands consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can
stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of
m
is
n
[IIT JEE Adv. 2015]
19.
If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL
and arranged as in a dictionary ; then the position of the word SMALL is :
[IIT JEE Main 2016]
th
th
th
th
(A) 46
(B) 59
(C) 52
(D) 58
20.
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club
including the selection of a captain (from among these 4 members) for the team. If the team has to include
at most one boy, them the number of ways of selecting the team is :
[IIT JEE Adv. 2016]
(A) 380
(B) 320
(C) 260
(D) 95
21.
Let S = {1, 2, 3,....9}. For k = 1, 2,....5. Let N k be the number of subsets of S, each
containing five elements out of which exactly k are odd. Then
[JEE Adv. 2017]
N1 + N2 + N3 + N4 + N5 =
(A) 126
22.
(C) 210
(D) 125
A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are
ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which
X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are
in this party, is :
[JEE Mains 2017]
(A) 485
23.
(B) 252
(B) 468
(C) 469
(D) 484
A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one – by – one, with
replacement, then the variance of the number of green balls drawn is :
(A)
24.
12
5
(B) 6
(C) 4
[JEE Mains 2017]
(D)
6
25
Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let x be the number of such
words where no letter is repeated ; and let y be the number of such words where exactly one letter is
repeated twice and no other letter is repeated. Then
25.
y
=
9x
[JEE Adv. 2017]
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and
arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements
is :
[JEE Mains 2018]
(A) at least 750 but less than 1000
(B) at least 1000
(C) less than 500
(D) at least 500 but less than 750
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
26.
In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and
5 girls G1, G2, G3, G4, G5.
(i) Let 1 be the total number of ways in which the committee can be formed such that the committee has
5 members, having exactly 3 body and 2 girls.
(ii) Let 2 be the total number of ways in which the committe can be formed such that the committee has
at least 2 members, and having an equal number of boys and girls.
(iii) Let 3 be the total number of ways in which the committe can be formed such that the committee has
5 members, at least 2 of them being girls.
(iv) Let 4 be the total number of ways in which the committee can be formed such that the commitee
has 4 members, having at least 2 girls and such that both M1 and G1 are NOT in the committee together.
LIST-I
LIST-II
P. The value of 1 is
1. 136
[JEE Adv. 2018]
Q. The value of 2 is
2. 189
R. The value of 3 is
3. 192
S. The value of 4 is
4. 200
5. 381
6. 461
The correct option is :(A) P 4; Q 6, R 2; S 1
(B) P 1; Q 4; R 2; S 3
(C) P 4; Q 6, R 5; S 2
(D) P 4; Q 2; R 3; S 1
27.
The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the
repetition of digits is allowed, is.
[JEE Adv. 2018]
28.
The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible
by 11 and no digit is repeated, is :
[JEE Main 2019]
(A) 36
(B) 60
(C) 48
(D) 72
29.
Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one
ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total
number of balls used in forming the equilaterial triangle, then all these balls can be arranged in a square
whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains.
Then the number of balls used to form the equilateral triangle is :
[JEE Main 2019]
(A) 190
(B) 262
(C) 225
(D) 157
30.
A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the
committee is formed with at least 6 males and n is the number of ways the committee is formed with at
least 3 females, then :
[JEE Main 2019]
(A) m = n = 78
31.
(B) n = m – 8
(C) m + n = 68
(D) m = n = 68
The number of four-digit numbers strictly greater than 4321 that can be formed using the digits
0, 1, 2, 3, 4, 5 (repetition of digits is allowed) is :
[JEE Main 2019]
(A) 288
(B) 306
(C) 360
(D) 310
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
32.
All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number
of such numbers in which the odd digits occupy even places is :
[JEE Main 2019]
(A) 175
(B) 162
(C) 160
(D) 180
33.
There are m men and two women participating in a chess tournament. Each participant plays two games
with every other participant. If the number of games played by the men between themselves exceeds the
number of games played between the men and the women by 84, then the value m is:
[JEE Main 2019]
(A) 12
(B) 11
(C) 9
(D) 7
34.
Consider three boxes, each containing 10 balls labelled 1, 2, ....., 10. Suppose one ball is randomly
drawn from each of the boxes. Denote by ni, the label of the ball drawn from the ith box, (i = 1, 2, 3 ).
Then, the number of ways in which the balls can be chosen such that n1 < n2 < n3 is: [JEE Main 2019]
(A) 120
(B) 82
(C) 240
(D) 164
35.
Let S = {1, 2, 3, ......, 100}. The number of non-empty subsets A of S such that the product of elements
in A is even is:
[JEE Main 2019]
(A) 2100 – 1
(B) 250 (250 – 1)
(C) 250 –1
(D) 250 + 1
36.
The number of functions f from {1, 2, 3, ....... , 20} onto {1, 2, 3, ..........., 20} such that f(k) is a multiple
of 3, whenever k is a multiple of 4, is :
[JEE Main 2019]
(A) 65 × (15)!
(B) 5! × 6!
(C) (15)! × 6!
(D) 56 × 15
37.
The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is:
[JEE Main 2019]
(A) 1256
(B) 1465
(C) 1365
(D) 1356
38.
The number of natural numbers less than 7,000 which can be formed by using the digits 0,1,3,7,9
(repitition of digits allowed) is equal to :
[JEE Main 2019]
(A) 250
39.
(C) 372
(D) 375
Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that
can be formed from this class, if there are two specific boys A and B, who refuse to be the members of
the same team, is :
[JEE Main 2019]
(A) 500
40.
(B) 374
(B) 200
(C) 300
(D) 350
Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two
vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then
the number of elements in the set S is :
[JEE Main 2019]
(A) 9
(B) 18
(C) 32
(D) 36
41.
Five person A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one
of three colours red, blue and green, then the number of ways of distributing the hats such that the person
seated in adjacent seats get different coloured hats is
[JEE (advanced) 2019]
42.
Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is
[JEE Main 2020]
6
(A) 5
(B) (1/2)(6!)
(C) 6!
(D) (5/2)(6!)
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
43.
The number of ordered pairs (r,k) for which 6Cr = (k2 – 3)Cr + 1, where k is an integer, is :
[JEE Main 2020]
(A) 4
(B) 6
(C) 2
(D) 3
44.
If a, b and c are the greatest values of 19Cp, 20Cq and 21Cr respectively, then :
(A)
a
b
c
11 22 42
(B)
a
b
c
10 11 42
(C)
a
b
c
11 22 21
(D)
[JEE Main 2020]
a
b
c
10 11 21
45.
An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which
4 marbles can be drawn so that at the most three of them are red is _____.
[JEE Main 2020]
46.
The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the
word 'EXAMINATION' is ________.
[JEE Main 2020]
47.
If Cr 25Cr and C0 + 5 C1 + 9 C2 + ......+ (101) C25=225 k, then k is equal to _________.
[JEE Main 2020]
48.
If the letters of the word ’MOTHER’ be permuted and all the words so formed (with or without meaning)
be listed as in a dictionary, then the position of the word ’MOTHER’ is.........
[JEE Main 2020]
49.
Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path.
Each pair of stations is connected by a straight track only. Further, each pair of nearest stations is
connected by blue line, whereas all remaining pairs of stations are connected by red line. If the number
of red lines is 99 times the number of blue lines, then the value of n is:
[JEE Main 2020]
(A) 201
(B) 199
(C) 101
(D) 200
50.
The value of (2.1P0 – 3. 2P1+ 4.3P2 – ..... up to 51th term) +(1! – 2! + 3! – ...... up to 51th term) is equal
to:
[JEE Main 2020]
(A) 1-51(51)!
(B) 1+(52)!
(C) 1
(D) 1+ (51)!
51.
The total number of 3–digit numbers, whose sum of digits is 10, is ____
52.
A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is
correct. The number of ways, in which a candidate answers all six questions such that exactly four of the
answers are correct, is ______
[JEE Main 2020]
53.
The number of words, with or without meaning, that can be formed by taking 4 letters at a time from
the letters of the word ’SYLLABUS’ such that two letters are distinct and two letters are alike, is.
[JEE Main 2020]
54.
There are 3 sections in a question paper and each section contains 5 questions. A candidate has to
answer a total of 5 questions, choosing at least one question from each section. Then the number of
ways, in which the candidate can choose the questions, is:
[JEE Main 2020]
(A) 2250
(B) 2255
(C) 1500
(D) 3000
55.
Two families with three members each and one family with four members are to be seated in a row. In
how many ways can they be seated so that the same family members are not separated?
[JEE Main 2020]
3
3
(A) 2! 3! 4!
(B) (3!) (4!)
(C) 3! (4!)
(D) (3!)2 (4!)
[JEE Main 2020]
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
56.
The number of words (with or without meaning) that can be formed from all the letters of the word
“LETTER” in which vowels never come together is______
[JEE Main 2020]
57.
If the number of five digit numbers with distinct digits and 2 at the 10th place is 336k, then k is equal to
[JEE Main 2020]
(A) 8
(B) 7
(C) 4
(D) 6
58.
An engineer is required to visit a factory for exactly four days during the first 15 days of every month and
it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways
in which such visits to the factory can be made by the engineer during 1 – 15 June 2021 is
[JEE (Advanced) 2020]
59.
In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a
way that each of these rooms contains at least one person and at most two persons. Then the number of
all possible ways in which this can be done is
[JEE (Advanced) 2020]
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Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
ANSWER KEY
DPP-1
Q.1
128
Q.2
120
Q.6
(i) 24; (ii) 576; (iii) 360
Q.8
(i) 840, (ii) 120, (iii) 400, (iv) 240, (v) 480, (vi) 40, (vii) 60, (viii) 240
Q.9
(i) 120, (ii) 40, (iii) 40, (iv) 80, (v) 20
47
Q.3
C
Q.4
738
Q.5
Q.7
(i) 468000 ; (ii) 421200
Q.12 (a) 34 ; (b) 24
150
Q.10 243 ways
Q.11
Q.13 90720
Q.14 C
Q.15 4500
Q.17 263
Q.18 720
Q.19
2652
Q.20 84
Q.21 999
Q.22 5·49
Q.23
2880
Q.24 1728
Q.16 1024
Q.25 Infinitely many
DPP-2
Q.1
A
Q.2
B
Q.3
A
Q.4
Q.8
A
Q.9
(a) 20, (b) 21, (c) 10 Q.10
C
Q.5
D
Q.6
D
Q.11
K = 17
Q.4
r=8
Q.5
Q.9
A
C
C
Q.7
D
Q.12 154
DPP-3
Q.1
967680
Q.2 (a) 60 (b) 107
Q.7
B
Q.8
Q.3
D
(a) 213564, (b) 267th
D
Q.6
D
Q.10 B
Q.11
A
Q.5
C
Q.6
64800
Q.11
A
Q.12 B
Q.5
C
Q.12 A
DPP-4
Q.1
(x 1)x 1
Q.2
C
Q.3
A
Q.4
Q.7
43200
Q.8
D
Q.9
D
Q.10 A
DPP-5
24C4
A
Q.2
C
Q.3
3150
Q.7
A
Q.8
B
Q.9
(i) 243 ; (ii) 1, 10, 40, 80, 80, 32
Q.12 D
Q.13 C
Q.14 B
Q.4
25C
Q.1
Q.15 B
5
Q.10 D
Q.6
B
Q.11
378
Q.16 528
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39
Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
DPP-6
Q.1
2940 Q.2
172800
Q.3
8C
Q.6
719
Q.7
C
Q.8
C
Q.11
B
Q.12 219
4
·4!
Q.13 D
Q.4
C
Q.5
B
Q.9
A
Q.10 B
Q.14 A
Q.15 C
Q.16 D
DPP-7
Q.1
B
Q.2
2n 1 1
Q.3
C
Q.4
C
Q.5
Q.6
B
Q.7
256
Q.8
C
Q.9
1680
Q.10 B
; Q.12
(13!)
(8!)(5!)
Q.13 6930
(9!)(5!)
Q.11
(2!)
3
Q.14 432
A
Q.15 10
DPP-8
Q.1
D
Q.2
C
Q.8
C
Q.9
n = 8 Q.10 240, 240, 255, 480
Q.14 A
Q.3
A
Q.4
B
Q.5
C
Q.6
A
Q.11
C
Q.12 A
Q.5
C
Q.6
C
Q.9
C
Q.7
A
Q.13 C
Q.15 (A) R; (B) Q; (C) S; (D) P
DPP-9
Q.1
D
Q.7
72, 78120
Q.10 D
Q.2
Q.11
C
A
Q.3
B
Q.8
16 · 10! or
Q.12 B
Q.4
B
10C
Q.13
3
· 3! · 2! · 8!
B
Q.14 B
Q.15 A
Q.5
A
Q.6
B, C
Q.12 B, C
Q.16 (20) · 8! Q.17 B
DPP-10
Q.1
D
Q.2
A
Q.3
C
Q.4
C
Q.8
D
Q.9
B
Q.10 A, B, D
Q.11
Q.14 B,C,D Q.15 B, D
Q.16 A, B, C, D
Q.17 C, D
Q.19 A, B, D
Q.20 (A) R; (B) S; (C) Q; (D) P
Q.21 (A) T; (B) R; (C) P; (D) Q; (E) S
Q.22
3003
Q.24 771 ways
Q.26
12870
Q.29
141
Q.25 330
Q.28 (a) 315 – 3 · 215 + 3; (b) 2250
D
Q.7
B
Q.13 B, C
Q.18 A, C, D
Q.23 n!(n – 1)! – 2(n – 1)!
Q.27 159
Q.30 (n + 1)n 1
Q.31 296
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40
Maths IIT-JEE ‘Best Approach’ (MC SIR)
Permutation & Combination
EXERCISE–I
(mC3 + nC3 + kC3)
4316527
420
2.
5.
8.
(a) 744; (b) 7704
43200
960
3.
6.
9.
13, 156
145
24C . 15C
2
3
10.
1106
11.
12! ;
11! . 4!
(3!) 4 2!
13.
52!
(13! ) 4
14.
17.
20.
23.
26.
29.
5400
30
576
120, 216, 210
2! 3! 8!
(ii) 792 ; (iv) r = 3
15.
18.
21.
24.
27.
45 C
16.
19.
22.
25.
28.
1638
2111
(a) 89, (b) 63
(i) 3359; (ii) 59; (iii) 359
30980
1.
4.
7.
m+n+kC
3
6
22100 , 52
4 . (4!)2 . 8C4 . 6C2
143
3888
;
52 !
3!(17 ! ) 3
EXERCISE–II
1.
4.
8.
11.
13.
15.
240 , 15552
2500
60
24
256
2252
2.
6.
9.
12.
14.
16.
510
3. 532770
27
7. 10, 3
64
10. 244
(i) 15, (ii) 126, (iii) 60, (iv) 105
(a) 72; 78120; (b) 23; (c) 32 ; (d) 435
440
17. 121
18.
n + 5C
19.
24!
23!
8C .
2
5 ;
1 (3!) 2 (2!) 5
(3!) (2!)
20.
23.
84
40
21.
24.
3119976
974
5
22. (a) 1680; (b) 1140
EXERCISE–III
1.
9.
14.
21.
28.
35.
42.
49.
56.
C
2.
B
3.
A
(A) P; (B) S; (C) Q; (D) Q
C
15.
7
16.
C
A
22.
A
23.
A
B
29.
A
30.
A
B
36.
C
37.
D
D
43.
A
44.
A
A
50.
B
51.
54
120 57.
A
58. 495.00
4.
10.
17.
24.
31.
38.
45.
52.
59.
B
6
C
11.
B
18.
5
25.
D
32.
B
39.
490
46.
135
53.
1080.00
C
D
5
B
D
C
2454
240
7.
12.
19.
26.
33.
40.
47.
54.
C
B
D
C
A
D
51
A
8.
13.
20.
27.
34.
41.
48.
55.
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C
C
A
625
A
30.00
309
B
41
