1. Accelerated Precalculus
Combinations
Accelerated
Precalculus

2. Review Problem
There are 21 people in this class. In how many different ways can we elect a President, a Vice-President, a Secretary and a Treasurer from this class?
Draw 4 blanks. 21 20 19 18 =

3. New Idea...
There are 21 people in this class. In how many different ways can we choose a committee of 4 to represent this class?
Draw 4 blanks.
But wait! The order we picked you in doesn’t matter this time. A committee of Autumn, Steven, Shail and Yash is the same as a committee of Yash, Autumn, Shail and Steven. So what do we do? 21 20 19 18
= ?

4. New Idea...
There are 21 people in this class. In how many different ways can we choose a committee of 4 to represent this class?
Draw 4 blanks.
Then divide by the number of ways we could arrange these four people! 21 20 19 18
= 5985 1 2 3 4

5. Combinations
A combination is an arrangement of objects in which order is NOT important!
Furthermore, the combination of n objects taken r at a time, written nCr or C(n, r) or is

6. Try These
5C3
= 10
8C4
= 70
5C2
= 10
= 6
6C5
7C0
= 1
6C1
= 6
7C7
= 1

7. Sample Problem #1
In how many different ways can I select 3 out of the 9 CD’s on my “wish list” to buy today?
Draw 3 blanks.
Then divide by the number of ways we could arrange these 3 CDs. 9 8 7
= 84 1 2 3

8. Sample Problem #2
There are 3 females and 7 males at math team practice. In how many different ways can 4 of them be chosen to be on the JV State team?
C(10, 4) = 10 9 8 7 1 2 3 4
= 210

9. Sample Problem #3
There are 3 females and 7 males at math team practice. In how many different ways can 4 of them be chosen to be on the JV State team, if exactly one is female?
Move the females to one room and the males to another... 3 1 7 6 5 1 2 3
= 105

10. Sample Problem #4 The sample space is C(10, 4) = 210
There are 3 females and 7 males at math team practice. If 4 of them are chosen to be on the JV State team, what is the probability that exactly two are female?
Remember the definition of probability…
The sample space is C(10, 4) = 210

11. Sample Problem #4
There are 3 females and 7 males at math team practice. If 4 of them are chosen to be on the JV State team, what is the probability that exactly two are female?
And the numerator is: C(3, 2)·C(7, 2) 3 2 1 7 6 1 2
= 63

12. Sample Problem #4
There are 3 females and 7 males at math team practice. If 4 of them are chosen to be on the JV State team, what is the probability that exactly two are female?
So the probability is:
210 63 3 10 =