Permutations and Combinations
A clothing store for men and boys has belts in five styles, and
there are seven sizes available in each style. How many
different kinds of belts does the store have?
s1, s2
t1, t2, t3
s1t1, s1t2, s1t3, s2t1, s2t2, s2t3
A room has six doors. In how many ways is it possible to enter
by one door and leave by another?
E1, E2, E3, E4, E5, E6
L1, L2, L3, L4, L5, L6
E1L2, E1L3, E1L4, E1L5, E1L6
E2L1,…
E3L1, …
E4L1, …
E5L1, …
E6L1, …
How many of the integers between 10,000 and 100,000 have no
digits other than 6, 7 or 8? How many have no digits other than
6, 7, 8, or 0?
3 x 3 x 3 x 3 x 3 = 35
003
03
3
30
3 x 4 x 4 x 4 x 4 = 3 x 44
In how many different orders can the four letters A, B, C, D be
written with no letter being repeated in any one arrangement?
4 x 3 x 2 x 1 = 24
A, B,
A (B,
B (A,
C (A,
D (A,
C,
C,
C,
B,
B,
D
D)
D)
D)
C)
AB (C, D)
AC (B, D)
AD (B, C)
ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
How many different license plates can be made if each plate has
three letters and repetition of letters on a license plate is
not allowed?
26 x 25 x 24
How many integers between 100 and 999 inclusive consist of
distinct odd digits?
5 x 4 x 3 = 60
How many integers between 100 and 999 inclusively are odd
numbers?
d1d2d3
9 x 10 x 5 = 450
1st digit is {1, 2, 3, 4, 5, 6, 7, 8, 9}
2nd digit is {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}
3rd {1, 3, 5, 7, 9}
How many integers are even numbers?
1st digit is {1, 2, 3, 4, 5, 6, 7, 8, 9}
2nd digit is {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}
3rd {0, 2, 4, 6, 8}
9 x 10 x 5 = 450
How many integers between 100 and 999 inclusively have distinct
digits? Of these how many are odd numbers?
d1d2d3
1st digit is {1, 2, 3, 4, 5, 6, 7, 8, 9}
2nd digit is {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}–{1st digit}
3rd digit is {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}–{1st digit,2nd digit}
9 x 9 x 8 = 648
648 numbers between 100 and 999 with distinct digits
How many odd numbers with distinct digits?
1st digit is {1, 2, 3, 4, 5, 6, 7, 8, 9} – { 1st digit and zero}
2nd digit is {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}–{1st digit, 2nd digit}
3rd digit is {1, 3, 5, 7, 9}
Start with the third digit and then the 1st digit
5 x 8 x 8 = 320 odd numbers
How many distinct even digits?
1st
1st
2nd
3rd
digit
digit
digit
digit
is
is
is
is
{1,
{1,
{1,
{0}
2,
2,
2,
or
3, 4, 5, 6, 7, 8, 9} – {3rd digit and zero} or
3, 4, 5, 6, 7, 8, 9} – {3rd digit
3, 4, 5, 6, 7, 8, 9, 0}
{2, 4, 6, 8}
Start with the third digit then the 1st digit
1 x 9 x 8 = 72 where 0 is the third digit
4 x 8 x 8 = 256 where 0 is not the third digit
328 even numbers
328 even numbers with distinct digits + 320 odd numbers with
distinct digits = 648 numbers with distinct digits
In how many ways can ten persons be seated in a row?
10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 10!
In how many ways can ten persons be seated in a row so that two
of them are not next to each other?
Look at a smaller version of the problem with only three people.
How many ways can you arrange A, B, C so that A and B do not sit
next to one another
ACB
BCA
ABC
ACB
BAC
BCA
CAB
CBA
XC
CX
X= AB or X = BA
ABC
CAB
BAC
CBA
With 3 people the formula is
3! – 2 x 2!
and with 10 people the formula is
2 x 9! ways two particular people can be seated next to one
another
10! ways they can be seated without restriction
10! – 2 x 9! Ways they can be seated apart from one another
An examination consists of ten questions, of which a student is
required to answer eight and omit two. In how many ways can a
student make his selection?
C(10, 8)
How many arrangements are there of the letters, taken all at a
time, of the word assesses?
hat
3!
hat
hta
aht
ath
tah
tha
haa
haa
aha
aah
ha(1)a(2)
ha(2)a(1)
a(1)ha(2)
a(2)ha(1)
a(1)a(2)h
a(2)a(1)h
8!/(2! x 5!)
How many different sums of money can be made up using one or
more coins selected from a cent, a nickel, a dime, a quarter, a
half dollar and a silver dollar?
C(6, 1) + C(6, 2) ... + C(6, 5) + C(6, 6) = 26 - 1
6 + 15 + 20 + 15 + 6 + 1 = 64 - 1
(1 + 1)6 = C(6,0)1610 + C(6,1)1511 + …
0 = (1 – 1)6
Do a simpler problem. How many different sum of money can be
made up using one or more coins selected from a cent, a nickel
and a dime
cent
nickel
dime
cent, nickel
cent, dime
nickel, dime
cent, nickel, dime
Probability
A teacher is going to separate ten boys into two teams of five
each to play basketball by drawing five names out of a hat
containing all ten names. As the drawing is about to start, one
boy says to a good friend, “I hope we get on the same team.” His
friend replies, “Well, we have a fifty-fifty change.” Is he
right, in the sense that the probability that the two boys will
be on the same team is ½?
C(10, 5)/2 ways to choose the two teams. We divide by 2 because
C(10, 5) defines pairs of teams and counts each pair twice (T1, T2) and (T2, T1)
C(8, 3) ways to choose two teams in which both boys are the same
team. We do not have to divide by 2 because whenever the boys
are on T1, the boys are not on the other team,T2, and (T1, T2) ≠
(T2, T1).
If ten coins fall to the floor, what is the probability that
there are five heads and five tails?
Does order matter?
No!
How many possible outcomes are there or what is the size of the
sample space.
HH
HT
TH
TT
2 x 2 = 4 = 22
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
2 x 2 x 2 = 8 = 23
...
2 x 2 x 2 ... x 2 = 210 = 1024
C(10, 5)
10!/(5!x5!)
P(5 heads) = C(10, 5)/ 210
What is graph?
What is a vertex?
What is an edge?
Graphs
Graphs represent communication systems
Graphs represent knowledge
What is a simple graph?
What is a loop?
What are parallel edges?
What is a complete graph?
What is a Bipartite graph?
What is the degree of a vertex?
Why is the does a loop add 2 to the degree of a vertex?
What is the total degree of a graph?
What is the relation between total degree of a graph and the
number of edges?
In any graph, how many vertices have odd degree?
Can you draw a graph with four vertices of degrees 1, 1, 2, and
3?
Can you draw a graph with four vertices of degrees 1, 1, 3, and
3?
Can you draw a simple graph with four vertices and degrees 1, 1,
3 and 3?
Is it possible in a group of nine people for each to be friends
with exactly five others?
Vegetarians and Cannibals
On a certain island the people are of two types – either
vegetarians or cannibals. Initially, two vegetarians and two
cannibals are on the left bank of a river. With them is a boat
that can hold a maximum of two people. The aim of the puzzle is
to find a way to transport all the vegetarians and cannibals to
the right bank of the river. What makes this so difficult is
that at no time can the number of cannibals on either bank
outnumber the number of vegetarians. Otherwise, disaster befalls
the vegetarians?
How do you get everybody on the right bank without anybody
eating anybody else? We will construct a graph whose vertices
are the various arrangements that can be reached in a sequence
of legal moves starting from the initial position. Connect
vertex x to vertex y if it is possible to reach vertex y in one
legal move from vertex x.