Sec 10.2 Permutations and Combinations
Permutation of a set of distinct objects is an
ordering of these objects
———————————–
The number of permutations of n objects is n!
read n factorial
————————————Permutations are handled in the same way as
sec 10.1 and the fundamental Th of counting
Problems # 1-40,permutations
Ex In how many ways can the letters abcd
be arranged
table:
1st letter 2nd letter 3rd letter 4th letter
4 ways
3ways
2
1
 4!  24 order is important
list: observe- (don’t copy)
abcd
bacd
cabd
dabc
abdc
badc
cadb
dacb
acbd
bcad
cbad dbac
acdb
bcda
cbda
dbca
adbc
bdac
cdab
dcab
adcb
bdca
cdba
dcba
formula:
P4, 4  4!  24
same answer
0!
—————————
Permutation of n things taken r at a time is
n!
written Pn, r, or P nr , or n P r
Pn, r 
n − r!
————————
do NOW
find
a. P8, 3
b. P9, 2
c. P5, 5
d. P5, 0
using n P r button on calculator
ans. a. 8!  336
b. 9!  72
5!
7!
5!
5!
c.
 120
d.
1
0!
5!
—————————
Test Question
1
Ex 1.A club has nine members . In how many ways can a
president, vice president and secretary be chosen from the
members of the club?
—————————–
hint: When specific officers are used, it is permutation
pres
vp sec
9 ways 8
7 ways
ans: 9  8  7  504
alternate: P9, 3  9!  504 ways
6!
Ex 2. From 20 raffle tickets in a hat, 4 tickets are selected.
The 1st ticket wins a car
2nd ticket wins a motorcycle
3rd ticket wins a bicycle
4th ticket, a skateboard
In how many different ways can these prizes be awarded?
—————————–
1st prize 2nd 3rd 4th
20 ways 19
18
17
ans.
20  19  18  17  116 280
alternate: P20, 4  20!  116 280 ways
16!
———————————–
do NOW p696
#19-25 odd
Distinguishable Permutations
If a set of n objects consists of k different kinds of objects with n 1 objects of the first
kind, n 2 objects of the second kind, n 3 objects of the third kind, and so on, where
n 1  n 2    n k  n, then the number of distinguishable permutations of these objects
is
n!
n 1 !n 2 !n 3 !n k !
——————————————Allows for repeated items
Ex 3. Find the number of different ways of placing 15 balls in a row
4 are red
3 are yellow
6 are black
2 are blue
—————————
Total of 15 balls n  15
2
red n 1  4
yellow n 2  3
black n 3  6
blue n 4  2
No. of different ways
repetitions:
n!
n 1 !n 2 !n 3 !n k !
 6, 306 , 300 ways
15!
4!3!6!2!
—————————Ex 4. Find the number of different ways of assigning 14
construction workers to 3 different tasks where
7 needed for mixing cement
5 for laying brick
2 for carrying bricks to layers
—————————
Total of 14 workers  n  14
repetitions:mix cement n 1  7
lay brick n 2  5
carry brick n 3  2
n!
No. of different ways
n 1 !n 2 !n 3 !n k !
14!  72 , 072 ways
7!5!2!
—————————–
CombinationsA combination of r elements of a set is any subset of r elements from the set (without
regard to order). If the set has n elements, then the number of combinations of r
elements is denoted by Cn, r or nr and is called the number of combinations of n
elements taken r at a time.
Combinations of n Objects Taken r At A Time
The number of combinations of n objects taken r at a time is
n!
Cn, r 
r!n − r!
note:Order is not important
—————————————
Ex In how many ways can the letters abcd
be arranged if order is not important
ans. one.
since abcd, and dcba are the same letters
in different order
—————————Test Question
Ex 5 A club has nine members. In how many ways can a committe of 3 be chosen
3
from the members of this club
————————
Note: since specific officers are not named, this is a combination
n!
Cn, r 
r!n − r!
n9, r3
C9, 3  9!  84
3!6!
——————————
do NOW
find
a. C8, 3
b. C9, 2
c. C5, 5
d. C5, 0
e. C10, 5
using n C r button on calculator
ans. a. 8!  56
b. 9!  36
3!5!
2!7!
5!
5!
c.
1
d.
1
e. 10!  252
5!0!
0!5!
5!5!
———————————Ex 6. From 20 raffle tickets in a hat, 4 tickets are selected.
The winners are awarded free trips to the Bahamas.
In how many different ways can these prizes be awarded?
—————————–
Since the order selected dosen’t matter,
this is a combination
ans: C20, 4  20!  4845 ways
4!16!
———————————–
#42.
Find the number of ways of selecting 3 pizza toppings, none repeated
from a set of 12 is
ans: order is not important
C12, 3  12!  220 ways
3!9!
———————————#46.
Find the number of ways of selecting 7 card hand from 52 cards
in a standard deck is
ans. order is not important
C52, 7  52!  133 , 784 , 560 ways
7!45!
———————————
# 56 There are 20 students, 12 female and 8 male.
4
Find the
a. Number of ways from 20 students for choosing a
committee of 5 is C20, 5  20!  15 , 504 ways
5!15!
b. Number of ways of choosing a committee of
5 females is C12, 5  12!  792 ways
5!7!
c. Number of ways of choosing a committee of
3 females and 2 males is
( no. of ways of choosing 3 from 12 females) 
( no. of ways of choosing 2 from 8 males)
C12, 3  C8, 2  22028  6160 ways
———————————————–
Read guidlines p694 for next topic
Test Question
Combining permutations and combinations
Ex 9 A committee of 7 consisting of a chairman, vice chairman
a secretary, and 4 other members is to be chosen from a class of 20 students
In how many ways can this committee be chosen?
———————————
committee consists of 2 parts:
a.selecting 3 named officers is P20, 3
b.This leaves 4 committee members from the 17 remaining
C17, 4
Total:
P20, 3andC17, 4  P20, 3  C17, 4
20!  17!  6840  2380  16 , 279 , 200 ways
17! 4!13!
—————————————Do NOW if time
p697 #41-61 odd
5