# How to express “$k$ things are the same” and “$n-k$ differ from each other as well as the $k$” as number of permutations?

How to express "$k$ things are the same" and "$n-k$ differ from each other as well as the $k$" as number of permutations?

I start with the obvious, which is the $n-k$ things differing from each other, which gives $(n-k)!$ permutations on the set "$n-k$".

However, I'm unsure about what counting the permutations of $n-k$ different things on the $k$ similar things means algebraically.

I believe $k$ similar things means that there's only one permutation, since with all orders of the $k$ elements, the sequence is the same.

But how do the two sets interact with each other? So what's the total amount of permutations?

Any hints?

• Have you talked about derangements? I can only assume that's what "$n - k$ differ from each other" would mean. – pjs36 Mar 1 '18 at 14:08

When $k$ things are the same, it means that you cannot distinguish between a permutation amongst them.
Assuming that all $n-k+k=n$ things would be different, there would be $n!$ permutations. However, since permutations of the indistinguishable things will result in the same overall permutation, one has to divide by the number of permutations under the indistinguishable things, which is $k!$. This gives $$\frac{n!}{k!}$$ overall permutations. Also refer to this website, and scroll down to "Permutations with Indistinguishable Objects".