Combinations

1

Permutations-Recall from yesterday

The number of possible arrangements (order matters) of a specific size from a group of objects is referred to as a permutation. The number of permutations of n elements taken r at a time is # in the group n P r  (

n n

 !

# taken from the group

r

)!

.

Example: You are required to read 5 books from a list of 8. In how many different orders can you do so?

n P r  P  8!



 8!

3!

=         

 6720 ways What if order does not matter? What if you are just concerned with which books are read? How will the numbers compare, why is there a difference?

2

Permutations

The number of possible arrangements (order matters) of a specific size from a group of objects is referred to as a permutation. Example 1: You are wish to select a president and vice preside nt from a group of four people. List all possibilities. (HINT call the four people A, B, C, and D.) Example 2: What if, you need to select two people from a group of four. List all the possibilities. Is this a permutation? How is this different?

3

Combination of n Objects Taken r at a Time

A combination is a selection of things when order does not matter. The number of combinations of r r objects from a group of objects selected from a group of n n objects is # in the collection n C

r

 (

n



n

!

# taken from the collection .

Example 3 You are required to read 5 books from a list of 8. In how many different ways can you do so if the order doesn’t matter?

:

8 C 5

= 8!

3!5!

=    3!5!

= 56 combinations

4

EXAMPLE 4

Ten students are running in a race. The first place winner receives a blue ribbon, the second place finisher receives a red ribbon and the third place finisher receives a white ribbon. How many different ribbon presentations are possible?

EXAMPLE 5

Ten students are running in a race. The first three finishers qualify for the finals. How many different groups could qualify for the finals?

How are these two examples different?

5

Example 6 : A state’s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies?

ANSWER

6

Example 7 : You need to field a dodge ball team. The team is required to have 6 players, consisting of 3 girls and 3 boys. There are 7 girls and 8 boys to choose from. How many different dodge ball teams can you organize?

ANSWER

7

Example 8 : In order to conduct an experiment, 4 subjects are randomly selected from a group of 20 subjects. How many different groups of four subjects are possible?

ANSWER

8

Example 9 : In Illinois the Lotto game is played by selecting 6 unique numbers between 1 and 52. If your six numbers match the six that are drawn, you are a winner. How many different tickets are possible?

ANSWER

9

Example 10 : You are playing a card game with a standard deck of 52 playing cards. You are dealt three cards. How many different hands are possible?

ANSWER

10