Accelerated Precalculus

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Presentation transcript:

Accelerated Precalculus Combinations Accelerated Precalculus

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 = 143640

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 = ?

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

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

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

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

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

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

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

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

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 =