query: Prove $\sum_{i=0}^n \binom{x + n - i}{n - i} \binom{y + i}{i} = \binom{x+y+n+1}{n}$.
positive_passages.docid: 82742
positive_passages.text: Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $
negative_passages.docid: 126312
negative_passages.text: How can you prove $\binom {2n}{2} = 2\binom{n}{2} + n^2$?
negative_passages.docid: 106965
negative_passages.text: How to algebraically prove $\binom{n+m}{2} = nm + \binom{n}{2} + \binom{m}{2}$?
negative_passages.docid: 13236
negative_passages.text: Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$
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: 143113
negative_passages.text: Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
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: 49771
negative_passages.text: Prove that $\sum\limits_{k=0}^r\binom{n+k}k=\binom{n+r+1}r$ using combinatoric arguments.
negative_passages.docid: 57418
negative_passages.text: How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$
negative_passages.docid: 65672
negative_passages.text: Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?
negative_passages.docid: 48200
negative_passages.text: Simplify $\sum_{i=0}^n (i+1)\binom ni$
negative_passages.docid: 129304
negative_passages.text: Alternative combinatorial proof for $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^r$
negative_passages.docid: 2885
negative_passages.text: Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
negative_passages.docid: 1158
negative_passages.text: How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
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: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.