Methods of Counting Methods of Counting OutcomesOutcomes

BUSA 2100, Section 4.1

Counting RulesCounting Rules

Counting rules provide a way to determine the number of possible outcomes for a situation without having to list or count them all.

The first counting rule is called the Multiplication Principle. It applies to outcomes for which order matters, i.e. order makes a difference.

Multiplication PrincipleMultiplication Principle

Multiplication Principle: The total number of outcomes for an ordered situation is the product of the number of outcomes for each part of the situation.

Example 1: How many different phone numbers are possible with the same area code?

(Suppose first digit cannot be a zero.)

Multiplication Principle, p. 2Multiplication Principle, p. 2 Does order make a difference?

Multiplication Principle, p. 3Multiplication Principle, p. 3 Example 2: How many different license

plates are possible using three numbers followed by three letters?

(Suppose zeros are not allowed and repetitions are not allowed for letters.)

Multiplication Principle, p. 4Multiplication Principle, p. 4

The Multiplication Principle is applicable whenever: (1) Order matters, i.e. objects in different orders represent different outcomes;

(2) Repetitions may or may not be allowed, depending upon the content of the problem.

PermutationsPermutations Definition: A permutation is an ordered

arrangement of distinct objects (repetitions are not allowed).

Example 1: How many ways can 5 people line up?

Lines are ordered arrangements and the same person can’t be chosen twice (no repetitions). So we use permutations.

Permutations, Page 2Permutations, Page 2

Permutation problems are done in the same way as Multiplication Principle problems.

Permutations are a special case of the Multiplication Principle.

In a permutation, the numbers occur in descending order.

Permutations, Page 3Permutations, Page 3 What is the symbol for the product of

the integers from 5 down to 1?

Ex. 2: How many ways can 3 people be selected from 7 people if the 1st person chosen is President, the 2nd is Vice President, and the 3rd is Secretary?

CombinationsCombinations

Definition: A combination is a selection of distinct objects for which order is not important (does not matter).

Example 1: How many different committees of 3 people can be chosen from 7 people?

Is order important?

Combinations, Page 2Combinations, Page 2 For convenience, refer to the 7 people

as A,B,C,D,E,F,G. Note that ABC, ACB, BAC, BCA, CAB, and CBA all refer to the same 3 people.

They represent six permutations, but only one combination.

Combinations, Page 3Combinations, Page 3

Summary: If order matters, use the Multip. Principle or permutations; if order doesn’t matter, use combinations.