query: How to prove this formula $ \sum_{k=2}^n k(k-1)\binom{k}{n} = n(n-1) \cdot 2^{n-2}$?
positive_passages.docid: 149216
positive_passages.text: Give a combinatorial proof to show the following for all integers $n \geq 2$.
negative_passages.docid: 143337
negative_passages.text: How to prove that $\sum_{i=0}^n 2^i\binom{2n-i}{n} = 4^n$.
negative_passages.docid: 10777
negative_passages.text: Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
negative_passages.docid: 49066
negative_passages.text: Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
negative_passages.docid: 6539
negative_passages.text: Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$?
negative_passages.docid: 116461
negative_passages.text: Prove the identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$ using a combinatorial interpretation of the positive and negative terms
negative_passages.docid: 44111
negative_passages.text: Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?
negative_passages.docid: 1258
negative_passages.text: Prove $3^n = \sum_{k=0}^n \binom {n} {k} 2^k$
negative_passages.docid: 16347
negative_passages.text: Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$
negative_passages.docid: 146200
negative_passages.text: proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
negative_passages.docid: 122006
negative_passages.text: Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly"
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 99249
negative_passages.text: Sum of product of binomial coefficients: $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n + k}{k} = (-1)^n$
negative_passages.docid: 46840
negative_passages.text: How to prove that $\sum_{r=0}^n\binom{n}{r}2^r=3^n$
negative_passages.docid: 110441
negative_passages.text: Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
negative_passages.docid: 14239
negative_passages.text: How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
negative_passages.docid: 32791
negative_passages.text: Proving this binomial identity $\sum_{k=0}^n {n+k \choose k} \frac{1}{2^{k}}= 2^{n}$