Question

An NFL coach sometimes uses a defense that utilizes 5 defensive linemen, 4 linebackers, and 2 defensive backs. His roster (the players he can choose from) includes 7 defensive linemen, 6 linebackers, and 6 defensive backs. How many different ways can the coach pick the 11 players to implement this particular defense

Answer 1

there are 4725 different ways to pick the 11 players

Explanation:

assuming that a player can only play from its position and not from the other 2 , then each defensive position is independent from the others , and since the order each player is chosen is not relevant we have that:

total combinations = possible combinations of linemen * possible combinations of linebackers * possible combinations of defensive backs = combinations of 5 linemen from 7 * combinations of 4 linebackers from 6 * combinations of 2 defensive backs from 6 = [ 7!/(5!*(7-5)!] * [ 6!/(4!*(6-4)!] * [ 6!/(2!*(6-2)!] = 7!/(2!*5!)*[ 6!/(4!*2!)]² = 4725 combinations

thus there are 4725 different ways to pick the 11 players