I have a sum (let's call it $p$):
$$p:= \frac{1}{n!}\sum_{i=2}^l (i-1)\frac{(n-k)!}{(n-k-i+2)!}(n-i)!$$
where $l, n, k$ are fixed positive integers, and $k \leq n$.
I'd like to either simplify or approximate $p$, for the purpose of simplifying or approximating the sum
$$\sum_{k=2}^n \binom{n}{k-1}(k-1)!\binom{n}{k}k! p^k$$
where $p$ is that nasty first sum we had.
I don't really know any approximation or numerical techniques, so any advice on how to simplify or approximate the first or second sum in such a way that the second sum would tend to $0$ as $n$ approaches infinity would be very much appreciated.