Counting, Permutations, &

Combinations

A counting problem asks “how many ways” some event can occur.

Ex. 1: How many three-letter codes are there using letters A, B, C, and D if no letter can be repeated?

• One way to solve is to list all possibilities.

Ex. 2: An experimental psychologist uses a sequence of two food rewards in an experiment regarding animal behavior. These two rewards are of three different varieties. How many different sequences of rewards are there if each variety can be used only once in each sequence?

Next slide

a

b

c

c

c

b

b

a

a

•Another way to solve is a factor tree where the number of end branches is your answer.

Fundamental Counting PrincipleSuppose that a certain procedure P can be broken into n successive ordered stages, S1, S2, . . . Sn, and suppose that

S1 can occur in r1 ways.

S2 can occur in r2 ways.

Sn can occur in rn ways.

Then the number of ways P can occur isnrrr 21

Ex. 2: An experimental psychologist uses a sequence of two food rewards in an experiment regarding animal behavior. These two rewards are of three different varieties. How many different sequences of rewards are there if each variety can be used only once in each sequence?

Using the fundamental counting principle:

X1st reward 2nd reward

3 2

Permutations

An r-permutation of a set of n elements is an ordered selection of r elements from the set of n elements

!!

rn

nPrn

! means factorialEx. 3! = 3∙2∙1

0! = 1

Ex. 1:How many three-letter codes are there using letters A, B, C, and D if no letter can be repeated?

Note: The order does matter

24!1

!434 P

Combinations

The number of combinations of n elements taken r at a time is

!!

!

rnr

nCrn

Where n & r are nonnegative integers & r < n

Order does NOT matter!

Ex. 3: How many committees of three can be selected from four people?

Use A, B, C, and D to represent the people

Note: Does the order matter?

4!1!3

!434 C

Ex. 4: How many ways can the 4 call letters of a radio station be arranged if the first letter must be W or K and no letters repeat?

600,27

2324252

Ex. 5: In how many ways can our class elect a president, vice-president, and secretary if no student can hold more than one office?

3Pn

Ex. 6: How many five-card hands are possible from a standard deck of cards?

960,598,2552 C

Ex. 7: Given the digits 5, 3, 6, 7, 8, and 9, how many 3-digit numbers can be made if the first digit must be a prime number? (can digits be repeated?)

60453 Think of these numbers as if they were on tiles,

like Scrabble. After you use a tile, you can’t

use it again.

Ex. 8: In how many ways can 9 horses place 1st, 2nd, or 3rd in a race?

50439 P

Ex. 9: Suppose there are 15 girls and 18 boys in a class. In how many ways can 2 girls and 2 boys be selected for a group project?

15C2 X 18C2 = 16,065