Prove $\sum_{i=0}^n {{n \choose i} \times 2^i} = 3^n $
23557
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
151757
Simplify $\sum_{i=0}^{N/2}{N-i\choose i}$
143337
How to prove that $\sum_{i=0}^n 2^i\binom{2n-i}{n} = 4^n$.
90519
Combinatorial proof of $\sum\limits_{k=0}^n {n \choose k}3^k=4^n$
58428
Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
47670
Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$
67527
Proof by Induction that $3^n ≥ 1+2^n$
4201
Prove $\sum\binom{n}{k}2^k = 3^n$ using the binomial theorem
72639
Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
16177
Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
17638
Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
7083
Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$
46840
How to prove that $\sum_{r=0}^n\binom{n}{r}2^r=3^n$
114277
Proving in Discrete Structures Problem that $2^N = \binom{N}{0} + \binom{N}{1} + \binom{N}{2} + \dots + \binom{N}{N}$
70128
Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
1139
Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
107397
Algebraic proof for the identity $\sum_{k=0}^m(-1)^{m-k}{{n \choose {k}}}= {{n-1}\choose m}$