query: Combinatorics summation proof involving binomial coefficients
positive_passages.docid: 154280
positive_passages.text: Show this equality (The factorial as an alternate sum with binomial coefficients).
negative_passages.docid: 143945
negative_passages.text: Summation of combination of binomial coefficient
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: 61483
negative_passages.text: Prove the following identity combinatorially
negative_passages.docid: 57986
negative_passages.text: Sum of products of combinatorials
negative_passages.docid: 91648
negative_passages.text: Binomial coefficient proof
negative_passages.docid: 94060
negative_passages.text: Sum of square binomial coefficients
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: 156383
negative_passages.text: Proving a combinatorial identity without double counting
negative_passages.docid: 10875
negative_passages.text: proof of binomial identity involving double sum
negative_passages.docid: 72026
negative_passages.text: Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $
negative_passages.docid: 71564
negative_passages.text: Combinatorial proof of identity involving central binomial coefficients
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 118372
negative_passages.text: Combination proof for $n(n+1)2^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}$
negative_passages.docid: 75910
negative_passages.text: Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
negative_passages.docid: 82872
negative_passages.text: Combinatorial proof involving partitions and generating functions
negative_passages.docid: 59638
negative_passages.text: How to prove this result on binomial coefficients?