difficulties in a proof with binomial theorem

Question

how can one prove, that:

$\sum_{k=0}^{n}\binom{2n+1}{k}=2^{2n} $

I was trying to use the binomial theorem, but I do have difficulties with 2n+1 in the binomial coefficient. Thank you

Answer 1

Use the fact that $$\sum_{k=0}^{r}\binom{r}{k}=2^r\;\;\text{ for any }r\geq0$$ together with the fact that $$\binom{r}{k}=\binom{r}{r-k}\;\;\text{ for any }r\geq 0,\; 0\leq k\leq r$$ (Hint: there are $2n+2$ numbers in the range $\{0,1,\ldots,2n+1\}$, and the $n+1$ numbers $\{0,1,\ldots,n\}$ are precisely the first half of them.)