I am stuck with a series that I want to simplify-
$${2n+1\choose 0} + {2n+1\choose 1} + {2n+1\choose 2} + \dots +{2n+1\choose n}$$
I think somehow the result $${n\choose 0} + {n\choose 1} + {n\choose 2} + \dots + {n\choose n} = 2^n$$ has to be used.
I can't wrap my head around how the above series can be manipulated to get the required series.
Let's use the fact that $\binom{2n+1}{j} = \binom{2n}{j}+\binom{2n}{j-1}$ After rewriting your series in this way we get $$\binom{2n+1}{0} + \binom{2n}{0} + \binom{2n}{1} + \binom{2n}{1} + \ldots+ \binom{2n}{n-1} + \binom{2n}{n-1} + \binom{2n}{n}$$ Also that's true that $\binom{n}{k} = \binom{n}{n-k}$ and $\binom{2n+1}{0} = \binom{2n}{0}$, so we can rewrite this as $$\binom{2n}{0} + \binom{2n}{2n} + \binom{2n}{1} + \binom{2n}{2n-1} + \ldots +\binom{2n}{n+1} + \binom{2n}{n-1} + \binom{2n}{n}$$ And we see that it's the following series: $$\sum_{j = 0}^{2n}\binom{2n}{j} = 2^{2n}$$