Lesson 9
Math 3201
Combinations
In contrast to permutations, combinations are an arrangement of objects
without regard to order.
A formula will be developed and applied in problem solving situations.
Combinations is a grouping of objects where order does not matter.
Which of the following is a permutation or a combination?


My fruit salad is a combination of apples, grapes and strawberries.
The combination to the safe was 4-7-2.
Whether it is a combination of “strawberries, grapes and apples” or “grapes,
apples and strawberries”, it is still a fruit salad.
Where 4-7-2 is the combination of a safe not 2-4-7 would not open the safe.
Order is important for the safe, but not important for the fruit salad.
Example:

In a lottery, six numbers from 1 to 49 are selected
(Lotto 6/49). A winning ticket must contain the same six
numbers but they may be in any order.
If order matters, determine the number of permutations.
49

49!
 10 068 347 520
49  6!
If order does not matter, why is it necessary to divide the value by 6!
(The number of ways of arranging the six selected numbers)? The
number of combination is
49

P6 
C6 
49!
49  48  47  46  45  44

 13 983 816
49  6!6!
6!
Why is the number of combinations less than the number of
permutations?
Generally, given a set of n objects taken r at a time, the number of
n!
combinations is n C r 
.
r!n  r !
The connection to permutation is n C r 
n
Pr
.
r!
An assignment consists of three questions (A, B, C) and students are
required to attempt two.
3
C2 
3!
3
2!3  2!
A
_
B
_
B
_
C
_
C
_
A
_
Two questions and order doesn’t matter, therefore, the combination is 3.
Example:
At a local ice-cream store, you can order a sundae with a choice
of toppings. There are three different sauces to choose from
(chocolate, strawberry, butterscotch) and four different dry
toppings (peanuts, smarties, M&M, sprinkles). When selecting
one sauce and one dry topping, how many different sundaes
could you order?
3
3!
4!

1!3  1! 1!4  1!
 3 4
C1  4 C1 
 12
We can see the same result using the tree diagram.
Chocolate
P
SM
Strawberries
MM
SP
P
SM
MM
Butterscotch
SP
P
SM
MM
SP
We can see the same result using the Fundamental Counting Principle.
3
X
______
Choice for sauces
Example:
4
______
Choice for toppings
A volleyball team has 12 players. How many ways can the coach
choose the starting line-up of 6 players?
12
12!
6!12  6 !
12  11  10  9  8  7

6  5  4  3  2 1
12
10 9 8

 11     7
6 2
5 3 4
 11  2  3  2  7
C6 
 924
There are 924 combinations for coach to choose the starting line-up of 6
players from 12 players.
Example:
If a committee of 8 people is to be formed from a pool of 13
people, but Mitchell and Lisa must be on the committee, how
many selections can be made?
11
11!
6!11  6!
11  10  9  8  7

5  4  3  2 1
10 9 8
 11 
  7
5 2 3 4
 11  3  2  7
C6 
 462
Homework: Page 110 #1-4