Section 5.5
Counting Techniques
Suppose the license plates in a state take the
form: NLLLNNN, where N’s are an integer 0-9
and L’s are letters A-Z.
How many unique license plates are possible?
(A)10+26+26+26+10+10+10
(B) 10 × 26 × 25 × 24 × 9 × 8 × 7
(C) (10 × 9 × 8 × 7) + (26 × 25 × 24)
(D) 10 × 26 × 26 × 26 × 10 × 10 × 10
Multiplication rule:
If a task consists of a sequence of choices in
which there are p selections for the first choice,
q selections for the second choice, r selections
for the third choice, etc., then task can be done
in
𝑝×𝑞×𝑟×⋯
different ways.
10 × 26 × 26 × 26 × 10 × 10 × 10 =
175,760,000
An airport shuttle bus driver needs to pick up 4
separate passengers: a,b,c,d. How many
different ways can the driver pick up the
passengers? For example: abcd, abdc, dcba, etc.
(A)10
(B) 16
(C) 24
(D)120
Solve the following:
(A)0
(B) 4
(C) 10
(D)24
4!=?
An IRS agent only has time to perform 3 audits,
yet has 6 people whom need to be audited.
How many ways can the agent schedule her 3
appointments? (e.g. abc, cba , abe, abf, eba, … )
(A)6
(B) 24
(C) 120
(D)720
Number of permutations of n Distinct
Objects Taken r at a Time
The number of arrangements of r objects chosen
from n objects, in which
1. the n objects are distinct,
2. repetition of objects is not allowed, and
3. order is important, is given by the formula
n!
n Pr 
n  r !
5-8
EXAMPLE
Betting on the Trifecta
In how many ways can horses in a 10-horse race
finish first, second, and third?
The 10 horses are distinct. Once a horse crosses the
finish line, that horse will not cross the finish line
again, and, in a race, order is important. We have a
permutation of 10 objects taken 3 at a time.
The top three horses can finish a 10-horse race in
10!
10! 10  9  8  7!


 10  9  8  720 ways
10 P3 
7!
10  3! 7!
5-9
Suppose an IRS agent has time to audit 3 people
out of a pool of 6 to audit. If order does not
matter, how many pools of 3 people can be
chosen? (abe=eab since order does not matter.)
(A)6
(B) 20
(C) 120
(D)720
These pools are called combinations.
A combination is a collection, without
regard to order, of n distinct objects
without repetition. The symbol nCr
represents the number of combinations
of n distinct objects taken r at a time.
5-11
Number of Combinations of n Distinct
Objects Taken r at a Time
The number of different arrangements of r
objects chosen from n objects, in which
1. the n objects are distinct,
2. repetition of objects is not allowed, and
3. order is not important, is given by the formula
n!
n Cr 
r!n  r !
5-12
EXAMPLE
Simple Random Samples
How many different simple random samples of size 4
can be obtained from a population whose size is 20?
The 20 individuals in the population are distinct. In
addition, the order in which individuals are selected is
unimportant. Thus, the number of simple random
samples of size 4 from a population of size 20 is a
combination of 20 objects taken 4 at a time.
Use combination with n = 20 and r = 4:
20!
20! 20 19 18 17 16! 116, 280



 4, 845
20 C 4 
4!20  4 ! 4!16!
4  3 2 116!
24
There are 4,845 different simple random samples of
size 4 from a population whose size is 20.
5-13
Suppose a car dealer has 3 car models to
arrange in a line for show. There are 3 cars of
model A, 2 cars of model B, and 1 car of model
C. How many ways can the cars be arranged
where the cars within a model are not distinct?
For example: AAABBC, ABACAB, etc.
(A)12
(B) 24
(C) 60
(D)720
Permutations with nondistinct items
The number of permutations of n objects of
which n1 are of one kind, n2 are of a second
kind, . . . , and nk are of a kth kind is given
by
𝑛!
𝑛1 ! × 𝑛2 ! × ⋯ × 𝑛𝑘 !
where n = n1 + n2 + … + nk.
EXAMPLE
Arranging Flags
How many different vertical arrangements are
there of 10 flags if 5 are white, 3 are blue, and 2 are
red?
We seek the number of permutations of 10 objects,
of which 5 are of one kind (white), 3 are of a second
kind (blue), and 2 are of a third kind (red).
Using Formula (3), we find that there are
10!
10  9  8  7  6  5!

 2,520 different
5!  3!  2!
5!  3!  2!
vertical
arrangements
5-16