Find the number of different ways of dividing $m n$ things into $n$ equal groups.

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Solution

Let us understand this by considering $^{'} m n^{'}$ number of alphabets, of which $^{'} n^{'}$ number of groups exist.
$∴$ No. of alphabets in each group
$= \frac{m n}{n} = m$
Now, if there are $^{'} m n^{'}$ number of alphabets, of which $^{'} m^{'}$ alphabets are identical and such identical groups are $^{'} n^{'}$ , then number of arrangements of these alphabets $= \frac{(m n)!}{m! \times m! \times m! . . . . n (t i m e s)}$
$= \frac{(m n)!}{{(m!)}^{n}}$
So, this is the number of distribution of $^{'} m n^{'}$ things into $^{'} n^{'}$ equal groups.
Now, division is just the number of selection and not the arrangements.
$∴$ No. of ways of dividing $= \frac{(m n)!}{{(m!)}^{n} n!}$

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Find the number of different ways of dividing mn things into n equal groups.

Find the number of different ways of dividing $m n$ things into $n$ equal groups.