Question

An NFL coach sometimes uses a defense that utilizes 3 defensive linemen, 4 linebackers, and 4 defensive backs. His roster (the players he can choose from) includes 8 defensive linemen, 7 linebackers, and 7 defensive backs.
How many different ways can the coach pick the 11 players to implement this particular defense? _________

Answer 1

Answer:

68,600

Step-by-step explanation:

The order of the players is not important. For example, a defensive line of Shaq Lawson, Ed Oliver and Jerry Hughes is the same as a defensive line of Ed Oliver, Shaq Lawson and Jerry Hughes. So we use the combinations formula to solve this question.

Combinations formula:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

Defensive Lineman:

3 from a set of 8. So

$C_{8,3} = \frac{8!}{3!5!} = 56$

56 combinations of defensive lineman

Linebackers:

4 from a set of 7. So

$C_{7,4} = \frac{7!}{4!3!} = 35$

35 combinations of linebackers

Defensive backs:

4 from a set of 7. So

$C_{7,4} = \frac{7!}{4!3!} = 35$

35 combinations of defensive backs

How many different ways can the coach pick the 11 players to implement this particular defense?

56*35*35 = 68,600

68,600 different ways can the coach pick the 11 players to implement this particular defense