How to simplify following binomial expansion?

Question

$\dbinom{2n+1}{n+1}+\dbinom{2n+1}{n+2}+....+\dbinom{2n+1}{2n }+\dbinom{2n+1}{2n+1}$?

my textbook's answer is $2^{2n}$.

Answer 1

Hint : Use $$\binom{2n+1}{0}+\binom{2n+1}{1}+\binom{2n+1}{2}+\binom{2n+1}{3}+...+\binom{2n+1}{2n+1}=2^{2n+1}$$ , which follows from the formula for $(a+b)^{2n+1}$ by setting $a=b=1$ and the symmetry-property $$\binom{2n+1}{k}=\binom{2n+1}{2n+1-k}$$ for $k=0,...,2n+1$