Chapter 13.2, Problem 62AYU

Basketball Teams On a basketball team of 12 players, 2 play only center, 3 play only guard, and the rest play forward (5 players on a team: 2 forwards, 2 guards, and 1 center). How many different teams are possible, assuming that it is not possible to distinguish a left guard from a right guard or a left forward from a right forward?

To determine

To find: How many different teams are possible?

Answer to Problem 62AYU

There are 126 possible ways of selecting the team.

Explanation of Solution

Given:

On a basketball team of 12 players, 2 only play center, 3 only play guard, and the rest play forward (5 players on a team: 2 forwards, 2 guards, and 1 center). How many different teams are possible, assuming that it is not possible to distinguish left and right guards and left and right forwards?

Formula used:

$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$

Calculation:

The team has 12 players, 2 only play center, 3 only play guard, and the rest play

forward (5 players on a team: 2 forwards, 2 guards, and 1 center).

Number of selections for guards $=C\left(3,2\right)$.

Number of selections for center $=C\left(2,1\right)$.

Number of selections for forwards $=C\left(7,2\right)$.

Hence, the number of different possible is $=C\left(2,1\right)*C\left(3,2\right)*C\left(7,2\right)$.

$=\frac{2!}{1!1!}*\frac{3!}{2!1!}*\frac{7!}{2!*5!}=126$

There are 126 possible ways of selecting the team.