query: Combinatorial proof of this question?
positive_passages.docid: 1158
positive_passages.text: How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
negative_passages.docid: 139842
negative_passages.text: Seeking a combinatorial proof $\sum\limits _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$
negative_passages.docid: 140416
negative_passages.text: Catalan number - combinatoric
negative_passages.docid: 97254
negative_passages.text: What is the proof of permutations of similar objects?
negative_passages.docid: 156566
negative_passages.text: Combinatorial proof of a binomial identity: $\sum_k \binom {2r} {2k-1}\binom{k-1}{s-1} = 2^{2r-2s+1}\binom{2r-s}{s-1}$
negative_passages.docid: 26343
negative_passages.text: Number of certain (0,1)-matrices, Stanley's Enumerative Combinatorics
negative_passages.docid: 152084
negative_passages.text: Binomial-Theorem proof
negative_passages.docid: 34299
negative_passages.text: Combinatorial proof of $\sum_{k=0}^n k \cdot k! = (n+1)! -1$
negative_passages.docid: 4610
negative_passages.text: Proving an identity with a combinatorial proof: $\binom{n}{k}\binom{k}{r}=\binom{n}{r}\binom{n-r}{k-r}$
negative_passages.docid: 67927
negative_passages.text: Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$
negative_passages.docid: 19571
negative_passages.text: Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
negative_passages.docid: 10917
negative_passages.text: Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
negative_passages.docid: 142978
negative_passages.text: combinatorics: sum of product of integer compositions
negative_passages.docid: 57986
negative_passages.text: Sum of products of combinatorials
negative_passages.docid: 103347
negative_passages.text: combinatorial proof of summation
negative_passages.docid: 39724
negative_passages.text: Combinatorial Identities
negative_passages.docid: 103538
negative_passages.text: Partitions of $n$ into distinct odd and even parts proof