The Student Room Group

S1 Permutation

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.


Answer = 9450
But Not sure how to get there!
Reply 1
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.
Reply 2
maybe we can only need to consider less cases since the cases of choosing from 6 badminton players are the same as choosing from 6 tennis players
Reply 3
I agree with the seven cases, but should it not be:
1*2*3 6 Choose 1 * 6 Choose 2 *5 Choose 3
1*3*2 etc.

I don't see how the number 9450 can be reach by adding, on the other hand a number in excess of 9450 is achieved by multiplying.
Reply 4
The answer is

[6 Choose 1 * 6 Choose 2 * 5 Choose 3]
+ [6 Choose 1 * 6 Choose 3 * 5 Choose 2]
+ ...