This is actually an identity of some kind. To prove it, I would recommend writing this out in factorial form. However, this is not true in all cases -- there is a very important condition that is missing from this statement that I state in the verification process.

So it follows that

Now when we're working with mods, , and this inverse exists iff (i.e. and ).

Supposing that exists, then it follows that

Thus, I leave it for you to finish verifying that given that .

I hope this makes sense!