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

68,600

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) = (n!)/(x!(n-x)!)`

Defensive Lineman:

3 from a set of 8. So

`C_(8,3) = (8!)/(3!5!) = 56`

56 combinations of defensive lineman

Linebackers:

4 from a set of 7. So

`C_(7,4) = (7!)/(4!3!) = 35`

35 combinations of linebackers

Defensive backs:

4 from a set of 7. So

`C_(7,4) = (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