Integral representation of simplified factorial division

Question

I know that $\int_0^\infty{t^ne^{-t}}dt=n!$ from the gamma function, but I'm looking for a single integral which can represent $\frac{n!}{(n-m)!}=n(n-1)(n-2)\cdots(n-m+1)$ so that I don't need to first calculate the factorial of some large number and then divide the result by the factorial of another large number. For example, the equation in the birthday problem $\frac{365!}{(365-n)!365^n}$ would be simplified by not computing $365!$ or $(365-n)!$, but only the resulting expression from simplifying their quotient.