I got stuck at the following summation while solving another problem.

$$\sum_{k=n}^N \frac{(k)!}{(k-n)!} $$

I expanded the summation but have no clue how to simplify it.


Let's rewrite original sum: $$ \sum_{ k=n} ^ N k!/(k-n)! = \sum_{k=0}^{N'} (k+n)!/k! = n! \sum_{k=0}^{N'} \frac {(k+n)!}{k!n!} = n! \sum_{k=0}^{N'} C_k^{n+k} = n! C^{n+N'+1}_{n+1} = n! C^{N+1}_{n+1} $$ ($N'=N-n$)

Second to last equality is easily proved with induction. (There might be some errors with limits of summation etc., I'm writing this on mobile, so it's hard to check).


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