Combinations & Permutations

Essentials: Permutations & Combinations(So that’s how we determine the number of possible samples!)

Definitions: Permutation; Factorial; Combination.

What a Factorial is and how to use it.

Ability to determine the number of permutations or combinations resulting from a stated situation.

Extras here: Tree diagrams & the multiplication rule.

Factorials The Factorial of a number is the multiplication of that

number by every smaller number down to 1.

The Factorial Notation is n!, where n represents the number and the “!” indicates the factorial process.

Note the following: By definition 0! = 1

Example: 8! = 8 * 7 *6 * 5 * 4 * 3 * 2 *1 = 40,320

Permutations

A Permutation is an arrangement of n objects in a specific order using r objects at a time.

)!(

!

rn

nPrn

Permutations: Examples

Example: A news program has time to present 2 of four available news stories. How many ways can the evening news be set up?

)!(

!

rn

nPrn

1*2

1*2*3*424 P

)!24(

!42 Pn

122

2424 P

Checking the process: If we let A, B, C, D represent the four shows, then the possible show orders would be:AB BA CA DAAC BC CB DBAD BD CD DC

Twelve (12) possible presentations where order matters.

Combinations

A combination is the selection of r objects from n objects without regard to order.

!)!(

!

rrn

nCrn

Combinations: Examples

!2)!24(

!42 Cn

1*2*1*2

1*2*3*42 Cn

Example: A news program has time to present 2 of four available news stories. How many different sets of stories can be presented on the evening news?

!)!(

!

rrn

nCrn

64

242 Cn

Checking the process: If we let A, B, C, D represent the four shows, then the possible show orders would be:AB BA CA DAAC BC CB DBAD BD CD DCHowever, AB and BA represent the presentation of the same two stories. If order does not matter, one of these two may be deleted. Repeating the process results in: AB, AC, AD, BC, BD, CD,

Six (6) different presentations where order does not matter.

Tree Diagrams

A Tree Diagram systematically lists all possible ways a sequence of events can occur.

Advantage: Visual display of sequential events.

Disadvantage: Only practical where the number of choices are small.

Example: What are the possible

results of flipping a coin twice?

HH T

HT T

Results: HH HT TH TT

Multiplication Rule (events independent)

In a sequence of n events in which the first event has k1 possibilities, the second event k2, etc. (to kn), the total number of possibilities is k1* k2* …* kn-1* kn. Example 1:

What are the possible results of flipping a coin twice? Answer: 2 * 2 = 4

The Multiplication Rule replaces tree diagrams of any size. Consider the tree diagram that would result from the following example. Example 2:

What are the dinner possibilities if there are 10 beverages, 6 appetizers, 11 entrees, and 8 desserts?

Answer: 10 * 6 * 11 * 8 = 5,280 dinner possibilities.