Why is choosing from the remainder incorrect in solving this combination question? There are 4 badminton, 5 volleyball and 6 swimmers. What are the number of ways to form a delegation of 4 players which has to have at least 1 player from each of the 3 sports.
My answer was $\binom{4}{1}\cdot\binom{5}{1}\cdot\binom{6}{1}\cdot\binom{12}{1}$. My reasoning was choosing one player from each sport and the remaining pool of players.
The correct answer however is $\binom{4}{2}\cdot\binom{5}{1}\cdot\binom{6}{1}+\binom{4}{1}\cdot\binom{5}{2}\cdot\binom{6}{1}+\binom{4}{1}\cdot\binom{5}{1}\cdot\binom{6}{2}$. Which gives a different numerical value. I understand the answer's line of reasoning of adding each of the 3 cases where the last player is chosen from a sport separately.
However I don't understand why my solution would give the incorrect answer.

You are double counting some teams. Consider two volley ball players $V_1$ and $V_2$. You can choose $V_1$ as part of 5C1 and $V_2$ as part of 12C1. Again $V_2$ can be selected as part of 5C1 and $V_1$ as part of 12C1. This counts this combination twice.
