Okay so I have 1000 houses in an area where lightning strikes one house at random every week, and I need to approximate the probablity of lightning striking the same house twice (or more, presumably) within one year (52 weeks), and then for two years. I'm allowed to use the Stirling Formula: $\displaystyle \sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}<n!<\sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n+\frac{1}{12(n-1)}}$.

Now if I did this right, with n multichoose k = $\displaystyle \left(\begin{array}{c} n+k-1\\

k\end{array}\right)$ I have $\displaystyle \left(\begin{array}{c} 1051\\

52\end{array}\right)$ possible outcomes with $\displaystyle \left(\begin{array}{c} 1051\\

52\end{array}\right)-\left(\begin{array}{c} 1000\\

52\end{array}\right)$ favorible outcomes which expanding with $\displaystyle \left(\begin{array}{c} n\\

k\end{array}\right)=\frac{n!}{(n-k)!k!}$ and then dividing leaves me with $\displaystyle \frac{1000!999!}{948!1051!}$(kind of messy). Is there some way to further simplify this before plugging it into Stirling, or perhaps a simpler way of going about this altogether?