2.5. Combinations http://www.ck12.org
Example C
There are 12 boys and 14 girls in Mrs. Cameron’s math class. Find the number of ways Mrs. Cameron can select a
team of 3 students from the class to work on a group project. The team must consist of 2 girls and 1 boy.
There are groups of both boys and girls to consider. From the 14 girls(n= 14 )in the class, we are choosing 2
(r= 2 ).
Girls:14 C 2 =14!
2!( 14 − 2 )!
14 C 2 =
14!
2!( 12 )!
14 C 2 =
87 , 178 , 291 , 200
2 ( 479 , 001 , 600 )
14 C 2 =
87 , 178 , 291 , 200
958 , 003 , 200
14 C 2 =^91
From the 12 boys(n= 12 )in the class, we are choosing 1(r= 1 ).
Boys:12 C 1 =12!
1!( 12 − 1 )!
12 C 1 =
12!
1!( 11 )!
12 C 1 =
479 , 001 , 600
1 ( 39 , 916 , 800 )
12 C 1 =
479 , 001 , 600
39 , 916 , 800
12 C 1 =^12
Therefore, the number of ways Mrs. Cameron can select a team of 3 students (2 girls and 1 boy) from the class of
26 students to work on a group project is:
Total combinations= 14 C 2 × 12 C 1 = 91 × 12 = 1 , 092
Points to Consider
- How does a permutation differ from a combination?
Guided Practice
There are 18 Democrats and 20 Republicans in a committee. Find the number of ways the committee can form a
sub-committee consisting of 3 Democrats and 4 Republicans.
Answer:
There are groups of both Democrats and Republicans to consider. From the 18 Democrats(n= 18 )in the committee,
we are choosing 3(r= 3 ).