Asymptotics for the sums from the inclusion-exclusion principle

Question

What is a method to compute the asymptotics of a sum resulting from the inclusion-exclusion principle? Each term of the sum can be approximated perhaps by Sterling's formula or the Gaussian distribution. However the alternating sign should effect some cancellation. As an example, the answer to this combinatorial problem generates a probability $$p=\frac c{n\choose j},$$ where $$c=\sum_{k=w}^j(-1)^{k-w}(j-k+1){n-k+1\choose j-k+1}.$$ What is the asymptote of $p$ for $\big|\frac jn-a\big|=o(n)$ and $\big|\frac wn-b\big|=o(n)$ for some positive numbers $a$ and $b$ as $n\rightarrow\infty$? Of course, the summands have alternating signs and the absolute values of which decreases, so we can always bound the sum with partial sums. But I am hoping for a neater expression for asymptotics and a general tool for producing it.