${1\over{n+1}} {2n \choose n} = {2n \choose n} - {2n \choose n-1}$ I need to prove that $${1\over{n+1}} {2n \choose n} = {2n \choose n} - {2n \choose n-1}$$ 
I started by writing out all the terms using the formula ${n!\over{k!(n-k)!}}$ but I can't make the two sides equal. 
Thanks for any help.

Multiply both sides by $n+1$. On RHS you get $\binom{2n}{n}(n-n+1)$ which will be equal to LHS. This is because 
$$
\binom{2n}{n-1}\cdot (n+1)=\binom{2n}{n-1}\cdot (n+1)\cdot \frac{n}{n}=n \cdot \binom{2n}{n}
$$
