# Permutations and combinations

• Oct 22nd 2008, 05:49 AM
Tangera
Permutations and combinations
Q: A school is asked to send a delegation of 6 players selected from 6 badminton players, 6 tennis players and 5 squash players. No pupil plays more than 1 game. The delegation is to consist of at least 1, and not more than 3, players drawn from each sport. Find the number of ways in which the delegation can be selected.

The answer I was given is 9450...but I don't know how to derive it...X.x Thank you for helping!
• Oct 22nd 2008, 07:39 AM
Plato
All I can give you is the coding for a computer algebra system.
$\sum\limits_{k = 1}^3 {\sum\limits_{j = 1}^3 {\sum\limits_{n = 1}^3 {\mbox{if}\left( {k + j + n = 6,{6 \choose k}{6 \choose j}{5 \choose n},0} \right)} } } = 9450$.
The “if” function returns the second argument if the first condition is true or else 0.

To do this otherwise, I think that it would necessary to list out the various options and calculate each case.
• Oct 22nd 2008, 11:36 AM
Soroban
Hello, Tangera!

Plato is absolutely correct.
Without his formula, a Brute Force listing is required.

Quote:

A school is asked to send a delegation of 6 players selected from:
. . 6 badminton players, 6 tennis players and 5 squash players.
No pupil plays more than 1 game.
The delegation is to consist of from 1 to 3 players from each sport.
Find the number of ways in which the delegation can be selected.

The answer I was given is 9450.

Let: . $\begin{array}{ccc}b &=& \text{no. of badminton players} \\ t &=& \text{no. of tennis players} \\ s &=& \text{no. of squash players} \end{array}$

There are seven possible cases . . .

$\begin{array}{ccc} (b,t,s) & \text{number of ways} \\ \hline (1,2,3) & \quad6{6\choose2}{5\choose3} \:=\:900 \\ \\[-4mm] (1,3,2) & \quad6{6\choose3}{5\choose2} \:=\:1200 \\ \\[-4mm] (2,1,3) & \quad{6\choose2}6{5\choose3} \:=\:900 \\ \\[-4mm] (2,3,1) & \quad{6\choose2}{6\choose3}5 \:=\:1500 \\ \\[-4mm] (3,1,2) & \quad{6\choose3}6{5\choose2} \:=\:1200 \\ \\[-4mm] (3,2,1) & \quad{6\choose3}{6\choose3}5 \:=\:1500 \\ \end{array}$
$\begin{array}{ccc}(2,2,2) & {6\choose2}{6\choose2}{5\choose2} \:=\:2250 \\ \\[-4mm] \hline & \qquad \text{Total: }\quad {\color{blue}9450}\end{array}$