Combinations and Permutations

What is a Combination?

A combination is an arrangement of items and events where order is not important. (order doesn’t matter)

e.g. rearranging a hand of cards that have been dealt doesn’t change the hand

What is a Permutation?

A permutation is an arrangement of items and events where each order of items counts as a different arrangement. (order matters)

e.g. a locker combination of 28-16-14 is a locker permutation, if it were in a different order the lock wouldn’t open

What is the difference?

whether order is important

the formula that is used

For ABC, all of the possible pairs if a different order counts as a different pair are AB, AC, BC, BA, CA, CB

Since AB and BA are the same 2 letters, you would cross out BA, CA, CB

Combinations and Permutations - Formulas

Calculating a combination - C

Calculating a permutation - P

Factorial Notationn! is read as “n factorial”

n! represents the number of ways that different objects can be selected to create ordered arrangements of size n

It’s calculated as n! = n X (n-1) X (n-2) X ...X 2 X 1

For example: If n = 5 then ((5 X (5-1) X (5-2) X (5-3) X (5-4))

5! = 5 X 4 X 3 X 2 X 1 = 120

Note: 0! = 1

Combinations - Examples

Example 1: There are 9 children that want to play a game that requires 3 players at a time. In how many ways can you choose a team of 3 children?

Example 2: From a class of 30 students, determine how many ways a five-person committee can be selected to organize a class party.

a) with no restrictions

b) with Marnie on the committee

Combinations - Examples

Example 3: The coach of a co-ed basketball team must select 5 players to start the game from a team that consists of 6 females and 5 males. How many ways can this be achieved if the coach must choose 3 females and 2 males to start the game?

Permutations - Examples

Example 1: There are 15 players on the school baseball team. How many ways can the coach complete the 9-person batting order?

Example 2: Determine the number of arrangements possible using the letters of the word MATHEMATICS.

Using the Graphing Calculator

Video -http://www.youtube.com/watch?v=eEC1f97XYGY

Homework

p.262 #1, 3, 4a,b, p.263, #7a-f

p.255 #1, 2, 3, p.256 #6, 7