Simplify $C(n,k)/C(n+1,k-1)$

Question

I need to simplify $C(n,k)/C(n+1,k-1)$ without the answer containing any factorials or binomial coefficients.
I know the answer is $((n-k+1) (n-k+2))/((n+1) k)$. I just have no clue how to go about getting there.

Answer 1

Your $x$ and $y$ should be $n$ and $k$, respectively.

Just expand the binomial coefficients into expressions that do involve factorials and then do a lot of cancelling:

$$\begin{align*} \frac{\binom{n}k}{\binom{n+1}{k-1}}&=\frac{\frac{n!}{k!(n-k)!}}{\frac{(n+1)}{(k-1)!((n+1)-(k-1))!}}\\\\ &=\frac{n!}{k!(n-k)!}\cdot\frac{(k-1)!(n-k+2)!}{(n+1)!}\\\\ &=\frac{n!}{(n+1)!}\cdot\frac{(k-1)!}{k!}\cdot\frac{(n-k+2)!}{(n-k)!}\\\\ &=\ldots\;? \end{align*}$$

Answer 2

Remember: ${n\choose k}=\frac{n!}{(n-k)!k!}$