Pemutations of n objects not all distinct

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SUMMARY

The formula for calculating the number of permutations of n objects, where p, q, and r are counts of indistinguishable objects, is given by n!/(p!q!r!). This formula arises from the need to account for the indistinguishability of the objects by dividing the total permutations n! by the factorials of the counts of each indistinguishable group (p!, q!, and r!). The reasoning is based on the principle that swapping indistinguishable objects does not create a new permutation, thus necessitating the division by the factorial of their counts.

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Suppose we want to give the number of permutations of n objects of which p are of one kind, q are of one kind and r are of one kind. Then the number of permutations are-
n!/(p!q!r!)
​
Can we somehow prove this how we got here?
 
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Imagine labeling each object so they ARE distinct. Then there are n! permutations. But if the labels are removed from, say, the p which "are of one kind", they are no longer distinct. That is, swapping the positions of two of those things would NOT be a different permutation. Since there are p such things, there are p! ways to swap only those things That is, of the n! ways of ordering those things, there are p! that are only rearranging those identical things- we need to divide by p! to count those as only one. Similarly for the "q" things of another kind and the "r" things of yet another:
[tex]\frac{n!}{p!q!r!}[/tex].
 
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