Transcript Section 5.5 - Humboldt State University
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: 4!=?
(A)0 (B) 4 (C) 10 (D)24
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
5-8 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 P r
n n
!
r
!
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
P
3 10 10!
3 !
10!
7!
10 9 8 7!
10 9 8 7!
720 ways 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
n C r
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 C r
r
!
n 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
C
4 20!
4 !
20!
4!16!
20 19 18 17 16!
4 3 2 1 16!
116,280 24 4,845 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
n
1 are of one kind,
n
2 kind, . . . , and
n k
are of a
k
th are of a second kind is given by 𝑛!
where
n
=
n
1 𝑛 1 ! × 𝑛 2 ! × ⋯ × 𝑛 𝑘 !
+
n
2 + … +
n k
.
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 5-16 10!
5!
3!
2!
10 9 8 7 6 5!
5!
3!
2!
2,520 different vertical arrangements