6.3
Find Probabilities Using Combinations
Example 1 Count combinations
Count the combination of two letters from
the list A, B, C, D, E.
Solution
List all the permutations of two letters in the list A,B, C, D, E.
Because order is important in a combination, cross out any
duplicate pairs.
AB AC AD AE BA BC BD BE CA CB
CD CE DA DB DC DE EA EB EC ED
There are ___
10 possible combinations of 2 letters from the list
A, B, C, D, E.
6.3
Find Probabilities Using Combinations
COMBINATIONS
Formula
The number of combinations of n objects
taken r at a time, where r < n, is given by:
n!
n Cr 
n  r ! r!
Example
The number of combinations of 5 objects taken 2
at a time is:
5!

5 C2 
5  2! 2!
5  4  3!
3!  21
 ___
10
6.3
Find Probabilities Using Combinations
Example 2 Use the combinations formula
Toppings You order a pizza at a restaurant. You
can choose 3 toppings from a list of 12. How
many combinations of toppings are possible?
Solution
The order in which you choose the toppings is not important.
So, to find the number of combinations of 12 toppings taken
3 at a time, find 12C3.
12!
12!

12 C 3 
12  3! 3!
9!  3!
12 1110  9!

9! 3  2 1
 220
___
There are _____
220
different
combinations of
toppings.
Find Probabilities Using Combinations
Checkpoint. Complete the following exercises.
6.3
1. Count the combination of two letters from the list A,
B, C, D, E, F.
6!
6 C2 
6  2! 2!

6  5  4!
4!  21
 ___
15
Find Probabilities Using Combinations
Checkpoint. Complete the following exercises.
6.3
2. In Example 2, suppose you can choose only
2 toppings out of 12 topping choices. How
many combinations are possible?
12!
12 C 2 
12  2! 2!
12 1110!

10! 21
 ___
66
6.3
Find Probabilities Using Combinations
Example 3 Find a probability using combinations
Scholarships A committee must award three students
with scholarships. Fifteen students are candidates for
the scholarship including you and your two best
friends. If the awardees are selected randomly, what is
the probability that you and your two best friends are
awarded the scholarships?
Solution
Write the number of possible outcomes as the number
Step 1
of combinations of 15 candidates taken 3 at a time, C .
15
3
15 14 13 12!
15!
15!


15 C 3 
15  3! 3! 12! 3! 12! 3  2 1
 455
___
6.3
Find Probabilities Using Combinations
Example 3 Find a probability using combinations
Scholarships A committee must award three students
with scholarships. Fifteen students are candidates for
the scholarship including you and your two best
friends. If the awardees are selected randomly, what is
the probability that you and your two best friends are
awarded the scholarships?
Solution
one of
Step 2 Find the number of favorable outcomes. Only ____
the possible combinations includes scholarships for you
and your two best friends.
Step 3 Calculate the probability.
1
P(scholarships awarded to you and your friends) = _____
455
Find Probabilities Using Combinations
Checkpoint. Complete the following exercises.
6.3
3. In Example 3, suppose there are 20
candidates for the scholarships. Find the
probability that you and your two best
friends are awarded the 3 scholarships.
20 19 18 17!
20!
20!


20 C3 
17! 3  2 1
20  3! 3!
17! 3!
 _____
1140
P(scholarships awarded to you and your friends) =
1
1140
6.3
Find Probabilities Using Combinations
Pg. 364, 6.3 #1-18