For a homogeneous Poisson process N on [0,$\infty$] show that for 0
$$P(N(s)=k \ | \ N(t))= {{N(t)}\choose{k}} (\frac{s}{t})^k (1-\frac{s}{t})^{N(t)-k} $$ for $k\leq N(t)$
N(t) is Poisson distributed as in the Cramér-lundberg model.
Is first recognised the result as the probability function for a binomial distribution. So what i want to find is probably P(X=k). where X $ \sim$ $Bin(\frac{s}{t},N(t))$. My problem is, I can't seem to find a result that gives the binomial. I can't find the link. I tried to write out the first, but all i get is something that is as well poisson distributed.