How many cricket teams can the manager choose if one particular batsman refuses to play when one particular bowler does? 
A certain country has a cricket squad of 16 people, consisting of 7 batsmen, 5 bowlers, 2 all- rounders and 2 wicket-keepers. The manager chooses a team of 11 players consisting of 5 batsmen, 4 bowlers, 1 all-rounder and 1 wicket keeper.  Find the number of different teams the manager can choose if one particular batsman refuses to be in the team when one particular bowler is in the team. 

I did it 
$6C5 \cdot 5C4 \cdot 2 \cdot 2 + 7C5 \cdot 4C3 \cdot 2 \cdot 2$. It is completely wrong. How to do it?

There are $5$ from $7 = 21$ ways to choose batsmen. For everyone of those there are $4$ from $5 = 5$ bowlers and for every one of those there are $2$ choices of wicket keeper and $2$ choices of all-rounders. Subtracting combinations that contain a certain bowler and batter involves counting the number of ways a batter can be in the group of $5$ which is $4$ from $6 = 15$, and for those, the choic of bowlers is $4$ from $4 = 1$. So the calculation looks like........
$([\binom{7}{5} - \binom{6}{4}] \cdot \binom{5}{4} + \binom{6}{4})\cdot 2 \cdot 2$ 
$=(6\cdot 5 + 15)\cdot 2 \cdot 2 =  180$
