Would like someone to double check my work here. Thank you :)
$|N|$ is the number of points of the unit rate Poisson point process up to time $t$. At time $t$, $|N| \sim Poisson(t)$. Therefore, $$ \begin{split} \mathbb{E} \left(|N| \choose k \right) &= \sum_{i=0}^\infty {i \choose k} \mathbb{P}(N = i) \\ &= \sum_{i=k}^\infty{i \choose k} \mathbb{P}(N = i) \\ &= \sum_{i=k}^\infty{i \choose k} \frac{e^{-t} t^i}{i!}\\ &=e^{-t} {1\over k!} t^k\sum_{i=k}^\infty \frac{t^{i-k}}{(i-k)!} \\ &= e^{-t} {1\over k!} t^k e^t \\ &= {t^k \over k!} \end{split} $$