how many ways we can select a team of $5$?

Question

There are $20$ A players and $6$ B Players, in how many ways a team of $5$ which includes at least $2$ Player A and at least $2$ Player B exists?

I found out it will be $ C_{20}^{2}\cdot C_{6}^{3}+C_{20}^{3}\cdot C_{6}^{2} $.

Answer 1

The team of five can be made up as the following....

$A, 2, 3$

$B, 3, 2$

The number of ways is therefore

$$\binom{20}{2}\cdot \binom{6}{3}+\binom{20}{3}\cdot\binom{6}{2}= 20900$$