252 Part 4: The State of the World
What if the top three vote-getters were chosen for the school board, instead of the top two?
There would be 8 v 7 v 6 = 336 permutations of eight people taken three at a time, but how
many combinations? If you were to list all the permutations, your list would include some
permutations that include Mr. Smith, Ms. Jones, and Dr. Johnson. How many permutations on
the list include these three people? That’s the same as asking how many permutations are there
of three people taken three at a time, and the answer is 3 v 2 v 1 = 6. Each group of three
people contributes 6 permutations to the total of 336. Because there are six arrangements of these
same three people—six arrangements of any three people—the number of permutations is six
times larger than the number of groups or combinations. Dividing the number of permutations
by 6, or 12 3
33P
P
, will give you the number of combinations. The combinations of n things taken t at a
time is nt nt
ttC P
Pn
ntt n
tnt!
!!!.
The permutations of 10 things taken 5 at a time is
10 510
10 510
5109876 54321
54321
P 30,240
!
!!
!
.The combinations of 10 things taken 5 at a time is10 5 10 5
5510
510 5109876 54321
54321 5
C P
P !
!! 432110 9 8 7 6
5432125222
.
The number of combinations will always be smaller than the number of permutations. Use per-
mutations when the arrangement or order is important and combinations when a group should
only be counted once, no matter how it’s arranged.CHECK POINT- Find the number of combinations of 6 things taken 3 at a time.
- Find the number of combinations of 7 things taken 4 at a time.
- Find the number of combinations of 5 things taken 2 at a time.
- How many different committees of 5 people can be chosen from a group of
12 people? - How many different ways can you order your ice cream sundae if you are going
to choose 3 toppings from a list of 12 possibilities and the order in which you
put toppings on does not matter?