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My book describes these methods as under:

The number of ways in which $m.n$ different things can be equally divided into $m$ groups is $$ \frac{(mn)!}{(n!)^{m} . m!}$$ .

And the number of ways $m.n$ different things can be equally distributed among $m$ distinct persons is $$ \frac{(mn)!}{(n!)^m}$$ .

My question is : Why does the formula later doesn't contain $m!$ but the former contain so? What is the reason?? The book reasons that

In case of distribution of things among the persons, the thing which has gone to one person shalln't go to the other unlike in case of division into groups and so is the formula.

I really couldn't understand this. Please explain me lucidly why these formulas are different and what the book wanted to tell. Please help.

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The groups are the same set of groups is the groups are permuted. There are $m$ groups and their permutations are $m!$. Permuting what each of $m$ persons will own is not the same distribution of good.

Visually: $\{A,B\}=\{B,A\}$. But $\text{Person }1\mapsto A$, $\text{Person }2\mapsto B$ is not the same as $\text{Person }1\mapsto B$, $\text{Person }2\mapsto A$.

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