The binomial formula. how to show: $\Sigma_{k=0}^n k \binom{n}{k} = n2^{n-1}$
1158
How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
89576
Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$
111098
Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
122006
Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
121661
Evaluating the sum $\Sigma \, \frac{1}{k(k+p)}$
40784
$\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
147910
Prove that $\sum_{n=0}^N\binom{2N-n}N2^n(n+1)=(1+2N)\binom{2N}N$
148949
Find the value of $\sum_{k=1}^{n} k \binom {n} {k}$
155769
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
64328
How to prove that the Binet formula gives the terms of the Fibonacci Sequence?
25004
Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically
18046
Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
60860
Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
56317
Intuitive explanation of binomial coefficient formula
118372
Combination proof for $n(n+1)2^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}$
37569
Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
52557
Derivation of binomial coefficient in binomial theorem.