combinatorial proof that $\sum_{i=0}^n {n+i\choose i}\frac{1}{2^i} = 2^n$
14239
How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
24588
Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
142132
How do you prove $\sum \frac {n}{2^n} = 2$?
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
35342
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
143113
Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
116356
Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$
146200
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
1158
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
4201
Prove $\sum\binom{n}{k}2^k = 3^n$ using the binomial theorem
103347
combinatorial proof of summation
74730
Combinatorial Proof of ${{n}\choose{k}} {{k}\choose{j}} = {{n}\choose{j}} {{n-j}\choose{k-j}}$
10211
Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
29828
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.