How do I prove that $\sum_{\ell\text{ odd}}^{n} \binom{n}{\ell} = 2^{n-1}$?
21152
Exactly half of the elements of $\mathcal{P}(A)$ are odd-sized
130223
Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$
4201
Prove $\sum\binom{n}{k}2^k = 3^n$ using the binomial theorem
14239
How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
45037
Show $(2^m-1,2^n+1)=1$ if $m$ is odd
10917
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
10777
Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
56828
A combinatorial proof that the alternating sum of binomial coefficients is zero
18046
Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
109576
Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$
75910
Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
92166
Number of odd binomial coefficients is a power of $2$
40784
$\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
108318
counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
1139
Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
146200
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$