WARM-UPEvaluate each expression.

1. 8! 2. 3. 9P514!

10!4!40,320 1001

15,120

Counting Techniques

Combinations

Combination

Counting technique in which order does NOT matter (i.e. CO is the same as OC).

Written nCrn=# of choicesr=arranged how many ways

!)!(!),(rrn

nrnC

Example 1: CombinationsEvaluate 10C4.

10C4 = 10! 4!(10 – 4)!

= 10! 4! • 6!

10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 14 • 3 • 2 • 1 • 6 • 5 • 4 • 3 • 2 • 1= / / / / / /

/ / / / / /

10 • 9 • 8 • 74 • 3 • 2 • 1=

= 210

Example 2: CombinationsEvaluate 12C3.

12C3 = 12! 3!(12 – 3)!

= 12! 3! • 9!

12*11*10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 13 • 2 • 1 •9*8*7* 6 • 5 • 4 • 3 • 2 • 1= / / / / / /

/ / / / / /

12 • 11 • 10 3 • 2 • 1=

= 220

/ / / / /

/

You Try!!!

Evaluate1)C(10,5)

252

2)C(8,2)28

3)C(25,7)480,700

Example 3

A reading list for a course in world literature has 20 books on it. In how many ways can you choose 4 books to read?

n=20 books to choose fromr= 4 books at a time

Find C(20,4)

= 4845

Example 4: CombinationsA disk jockey wants to select 5 songs from a new CD that

contains 12 songs. How many 5-song selections are possible?

Relate: 12 songs chosen 5 songs at a timeDefine: Let n = total number of songs.

Let r = number of songs chosen at a time.

Write: nCr = 12C5

You can choose five songs in 792 ways.

Use the nCr feature of your calculator.

YOU TRY!!!

In this class we have 20 people. In how many ways can I select 6 students?

38760