Combinations
1
Permutations-Recall from yesterday
The number of possible arrangements (order matters) of a
specific size from a group of objects is referred to as a
permutation.
The number of permutations of n elements taken r at
a time is
n!
n
# in the
group
Pr 
(n  r)!
.
# taken from
the group
Example: You are required to read 5 books from a list of 8.
In how many different orders can you do so?
8!  8! = 8  7  6  5  4  3  2  1  6720 ways
P

P

n r
8 5
(8  5)! 3!
3  2 1
What if order does not matter? What if you are just concerned
with which books are read? How will the numbers compare,
why is there a difference?
2
Permutations
The number of possible arrangements (order matters) of a
specific size from a group of objects is referred to as a
permutation.
Example 1: You are wish to select a president and vice preside
nt from a group of four people. List all possibilities. (HINT call
the four people A, B, C, and D.)
Example 2: What if, you need to select two people from a group
of four. List all the possibilities. Is this a permutation? How is
this different?
3
Combination of n Objects Taken r at a Time
A combination is a selection of r objects from a group of n
things when order does not matter. The number of
combinations of r objects selected from a group of n objects is
# in the
collection
nC r 
n!
.
(n  r)! r !
# taken from
the collection
Example 3:
You are required to read 5 books from a list of 8. In how
many different ways can you do so if the order doesn’t
matter?
8! = 8  7  6  5!
C
=
8 5
3!5!
3!5!
= 56 combinations
4
EXAMPLE 4
Ten students are running in a race. The first place winner
receives a blue ribbon, the second place finisher receives a
red ribbon and the third place finisher receives a white
ribbon. How many different ribbon presentations are
possible?
EXAMPLE 5
Ten students are running in a race. The first three finishers
qualify for the finals. How many different groups could
qualify for the finals?
How are these two examples different?
5
Example 6:
A state’s department of transportation plans to develop a
new section of interstate highway and receives 16 bids for
the project. The state plans to hire four of the bidding
companies. How many different combinations of four
companies can be selected from the 16 bidding companies?
ANSWER
6
Example 7:
You need to field a dodge ball team. The team is required
to have 6 players, consisting of 3 girls and 3 boys. There
are 7 girls and 8 boys to choose from. How many different
dodge ball teams can you organize?
ANSWER
7
Example 8:
In order to conduct an experiment, 4 subjects are randomly
selected from a group of 20 subjects. How many different
groups of four subjects are possible?
ANSWER
8
Example 9:
In Illinois the Lotto game is played by selecting 6 unique
numbers between 1 and 52. If your six numbers match the
six that are drawn, you are a winner. How many different
tickets are possible?
ANSWER
9
Example 10:
You are playing a card game with a standard deck of 52
playing cards. You are dealt three cards. How many
different hands are possible?
ANSWER
10