Combinations

Accelerated Math II November 17, 2014

Review Problem

• Suppose there are 21 members of Mu Alpha Theta. In how many different ways can we elect a Captain, a Co-Captain, a Secretary and a Treasurer from this group?

• Draw 4 blanks.

21 20 19 18 = 143640

= ?

New Idea...

• Suppose there are 21 people in Mu Alpha Theta. In how many different ways can we choose a team of 4 from this group?

• Draw 4 blanks.

21 20 19 18

But wait! The order we picked you in doesn’t matter this time. A team of Riya, Anastasia, Radhesh and Yash is the same as a team of Yash, Riya, Radhesh and Anastasia. So what do we do?

= 5985

New Idea...

• Suppose there are 21 people in Mu Alpha Theta. In how many different ways can we choose a team of 4 from this group?

• Draw 4 blanks.• Then divide by the number of ways

we could arrange these four people!21 20 19 181 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 isn!

(n r )! r !

n

r

Try These

5C3

5C2

7C0

7C7

= 10

= 10

= 1

= 1

6C5

8C4

6C1 = 6

= 6

= 70

= 84

Sample Problem #1• In how many different ways can I

select 3 out of the 9 pumpkins left at Kroger to buy today?

• Draw 3 blanks. • Then divide by the number of ways

we could arrange these 3 pumpkins.

9 8 71 2 3

= 210

Sample Problem #2• There are 3 ghosts and 7 zombies at

a Halloween party. In how many different ways can 4 of them be chosen to scare the other guests?

•C(10, 4) =

10 9 8 71 2 3 4

= 105

Sample Problem #3• There are 3 ghosts and 7 zombies at

a Halloween party. In how many different ways can 4 of them be chosen to scare new guests if exactly 1 is a ghost?

• Move the ghosts to one room and the zombies to another...

31

7 6 51 2 3

Sample Problem #4

• There are 3 ghosts and 7 zombies at a Halloween party. If we select 4 of them chosen at random to scare others, what is the probability that exactly two are ghosts?

• Remember the definition of probability…

•The sample space is C(10, 4) = 210

= 63

Sample Problem #4

• There are 3 ghosts and 7 zombies at a Halloween party. . If we select 4 of them chosen at random to scare others, what is the probability that exactly two are ghosts?

• And the numerator is: C(3, 2)·C(7, 2)

3 21 2

7 61 2

• There are 3 ghosts and 7 zombies at a Halloween party. If we select 4 of them chosen at random to scare others, what is the probability that exactly two are ghosts?

• So the probability is: =

Sample Problem #4

210

63 310

Assignment

• Page 647: 1-9 all, 17-33 odd, 34-43 all

•Good luck!