# How many ways to select a delegation from different sports players

A school is asked to send a delegation of six pupils selected from six badminton players, six tennis players and five squash players. No pupil plays more than one game. The delegation is to consist of at least one, and not more than three, players drawn from each game. Find the number of ways in which the delegation can be selected.

I use the following approach, but I failed to get the correct answer.

Split the problem into seven cases:

(number of badminton players) + (number of tennis players) + (number of squash players)
1 + 2 + 3
1 + 3 + 2
2 + 1 + 3
2 + 2 + 2
2 + 3 + 1
3 + 1 + 2
3 + 2 + 1


Find the number of possible selections for each case. Add the answers, gives 10350.

But the correct answer is 9450.

• So far your approach looks correct, but you might be computing the number of possible selections for a case wrong. Anyway, this is more combinatorics than probability, and might be better suited for math.SE (ask for the moderators to migrate rather than post a copy on our own.) – Juho Kokkala Apr 8 '14 at 10:05
• Is this a homework problem? If so, it should have the self-study tag. – Peter Flom - Reinstate Monica Apr 8 '14 at 10:17
• No this is not a homework problem. Just a practice problem – user42268 Apr 8 '14 at 10:23
• No. All the combinations should be calculated correctly – user42268 Apr 8 '14 at 10:27
• I get 9450 also. You need to show the number of possible selections you found for each case, and the general rule you used to get them. – TooTone Apr 8 '14 at 23:33

$= b+ t + s$
Then for each case you have ${6 \choose b}{6 \choose t}{5 \choose s}$ combinations.