First slide

Combinations

Question

The total number of ways of dividing $m n$ distinct objects into n equal groups is

Moderate

A

$\frac{(m n)!}{(m!)^{n} (n!)}$
B

$\frac{(m n)!}{(n!)^{m} (m!)}$
C

$\frac{(m n)!}{m! n!}$
D

none of these

Solution

As groups are indistinguishable, the number of ways
$\begin{array}{l} = \frac{1}{n!} (^{m n} C_{m}) (^{m n - m} C_{m}) \dots (^{m} C_{m}) \\ = \frac{1}{n!} \frac{(m n)!}{m! (m n - m)!} \frac{(m n - m)!}{m! (m n - 2 m)!} \dots \frac{m!}{m! 0!} \\ = \frac{1}{n!} \frac{(m n)!}{(m!)^{n}} \end{array}$

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