Binomial coefficients algebra

Question

I am trying to prove $$\binom{n+1}{k} = \binom{n+1}{k-1}\frac{n-k+2}{k}$$ by using the following four equations:
\begin{align*} \binom{n + 1}{k} & = \binom{n}{k} + \binom{n}{k - 1}\\ \binom{n + 1}{k - 1} & = \binom{n}{k - 1} + \binom{n}{k - 2}\\ \binom{n}{k} & = \binom{n}{k - 1}\frac{n - k + 1}{k}\\ \binom{n}{k - 1} & = \binom{n}{k - 2}\frac{n - k + 2}{k - 1} \end{align*} Assume the above equations are true and valid to use.

Answer 1

If in your last relation $$\binom{n}{k - 1} = \binom{n}{k - 2}\frac{n - k + 2}{k - 1}$$ you substitute $n$ with $n+1$ and $k$ with $k+1$, then you get the desired relation.

Answer 2

Use the third equation, replacing $n$ by $n+1$.

Answer 3

For beginners to these kinds of proofs, sometimes it is just easier renaming the variables. For example:

I am trying to prove $$\binom{n+1}{k} = \binom{n+1}{k-1}\frac{n-k+2}{k}$$ Assuming that the following is true $$\binom{r}{s}=\binom{r}{s-1}\frac{r-s+1}{s}$$ Notice that this is just your third identity, expressed in $r,s$ instead of $n,k$. Now it's just a simple matter of replacing $r$ with $n+1$ and $s$ with $k$.