How do i reduce this expression of binomial coefficients

Solution 1:

There is a variation of the Vandermonde identity that reads, for $k,m,n\in\mathbf N$: $$ \sum_{i=0}^k\binom im\binom{k-i}n=\binom{k+1}{m+n+1}. $$ Here is how you can remember it: let $0\leq a_0<\cdots<a_{m+n}\leq k$ be one of the $\binom{k+1}{m+n+1}$ subsets of $m+n+1$ numbers $a_j$ from the $k+1$-set $\{0,\ldots,k\}$, arranged increasingly. Put $i=a_m$, then there are $\binom im$ choices left for $a_0,\ldots,a_{m-1}$, and $\binom{k-i}n$ choices for $a_{m+1},\ldots,a_{m+n}$.

One can restrict the range of $i$ to the values $m\leq i\leq k-n$, as other terms contribute $0$.

So your expression simplfies to $$ \frac{{\sum\limits_{x=Q}^{N-P+Q} \binom{x-1}{Q} \binom{N-x}{P-Q}}}{{\sum\limits_{x=Q}^{N-P+Q} \binom{x}{Q} \binom{N-x}{P-Q}}}= \frac{\binom{N}{P+1}}{\binom{N+1}{P+1}}=\frac{N-P}{N+1}. $$