Proving $\sum_{k = 0}^n k\binom nk^2 = n \binom{2n-1}{n-1}$ with combinatorial proof
120976
Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
90904
Combinatorial proof of an identity with squared binomial coefficients
65672
Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?
92704
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
156566
Combinatorial proof of a binomial identity: $\sum_k \binom {2r} {2k-1}\binom{k-1}{s-1} = 2^{2r-2s+1}\binom{2r-s}{s-1}$
146200
proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
34299
Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
3786
Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$
111098
Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
72026
Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
118372
Combination proof for $n(n+1)2^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}$
70128
Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
30672
Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
113321
Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
11612
How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
21000
Calculating $\sum_{k=1}^nk(k!)$ combinatorially
103347
combinatorial proof of summation