Question

A basketball team has three pure centers, four pure forwards, four pure guards and one swing- man who can play either guard or forward. A "pure" position player can play only the designated position. If the coach must start a lineup with one center, two forwards and two guards, how many possible lineups can the coach choose?

Answer 1

Answer:

We conclude that he coach can choose  252  lineups.

Step-by-step explanation:

We have three pure centers, four pure forwards, four pure guards and one swing- man who can play either guard or forward.

First case: swing- man play guard.

So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 2, and from 4 from the guards to choose 1.

$C_1^3\cdot C_2^4\cdot C_1^4=3\cdot \frac{4!}{2!(4-2)!}\cdot 4=12\cdot 6=72$

Second  case: swing- man play forward.

So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 1, and from 4 from the guards to choose 2.

$C_1^3\cdot C_1^4\cdot C_2^4=3\cdot 4\cdot \frac{4!}{2!(4-2)!}=12\cdot 6=72$

Third case: swing- man not play.

So now we count the number of combinations: to select from 3 centers 1 center, from 4 forwards to choose 2, and from 4 from the guards to choose 2.

$C_1^3\cdot C_2^4\cdot C_2^4=3\cdot \frac{4!}{2!(4-2)!}\cdot \frac{4!}{2!(4-2)!}=3\cdot 6 \cdot 6=108$

Now, we get 72+72+108=252.

We conclude that he coach can choose  252  lineups.