Proving $\sum_{k=0}^{n} {n \choose k} = 2^n$ with Newton's Binomial Theorem
23557
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
44621
Simple binomial theorem proof: $\sum_{j=0}^{k} \binom{a+j}j = \binom{a+k+1}k$
114911
Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$
123955
How to compute $\sum\limits_{k=0}^n (-1)^k{2n-k\choose k}$?
35381
Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $
139842
Seeking a combinatorial proof $\sum\limits _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
81729
Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion
25826
Proof of a combinatorial identity: $\sum\limits_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$
136580
Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$
28027
Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
46840
How to prove that $\sum_{r=0}^n\binom{n}{r}2^r=3^n$
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
96903
How to show via inductive proof (and without the use of the binomial theorem) that $ {n\choose k} \in \mathbb{N} $?
59252
Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
156564
Prove the identity $\sum_{{{\underset{k-even}{k=0}}}}^{n}{n \choose k}2^{k}=\frac{3^{n}+(-1)^{n}}{2}$