Transcript 5. Permutations and Combinations

Methods of Counting
Outcomes
BUSA 2100, Section 4.1
Counting 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 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. 2

Does order make a difference?
Multiplication 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. 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.

Permutations
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 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 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?
Combinations



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 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 3

Summary: If order matters, use the
Multip. Principle or permutations; if
order doesn’t matter, use combinations.