Transcript Combinations Halloween

Combinations
Accelerated Math II
October 31, 2013
Review Problem
• Suppose there are 21 members of Mu
Alpha Theta. In how many different
ways can we elect a Captain, a CoCaptain, 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.
But wait! The order we picked you in
doesn’t matter this time. A team of Avery,
Jillian, Stephen and Chris is the same as a
20Avery,
19Stephen
?
18 and= Jillian.
team21
of Chris,
So what do we do?
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
1
20
2
19
3
18
4
= 5985
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
n

nCr or C(n, r) or   is
r
n!
(n  r) !  r!
Try These
5C3
= 10
5C2 = 10
8C4
= 70
7C0
= 1
=
6
C
6 5
7C7
= 1
6C1
= 6
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
1
8
2
7
3
= 84
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 be in a
picture?
• C(10, 4) =
10
1
9
2
8
3
7
4
= 210
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...
3
1
7
1
6
2
5
3
= 105
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
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
1
2
2
7
1
6
2
= 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?
• So the probability is:
3
=
10
210
63
Assignment
• Page 647: 1-9 all, 17-33
odd, 34-43 all
•Good luck!