query: Combinatorial Proofs with Summation on One Side
positive_passages.docid: 28936
positive_passages.text: Proving equation with binomial coefficient
negative_passages.docid: 68882
negative_passages.text: Path to Combinatorics Problem [$1.11$]
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: 65983
negative_passages.text: Combinatorial proof of binomial coefficient summation identity
negative_passages.docid: 92359
negative_passages.text: Big-O Notation Proof
negative_passages.docid: 115686
negative_passages.text: A finite summation of double binomial coefficients
negative_passages.docid: 57986
negative_passages.text: Sum of products of combinatorials
negative_passages.docid: 17580
negative_passages.text: Combinatorics That Looks Similar to Vandermonde's Identity
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: 28109
negative_passages.text: Conjectural closed-form representations of sums, products or integrals
negative_passages.docid: 146867
negative_passages.text: Combinatorial Proof Of A Number Theory Theorem--Confusion
negative_passages.docid: 145900
negative_passages.text: Proof of a double summation reduction identity
negative_passages.docid: 81729
negative_passages.text: Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion
negative_passages.docid: 158508
negative_passages.text: Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$
negative_passages.docid: 14986
negative_passages.text: Proving a combinatorics equality: $\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}$
negative_passages.docid: 83229
negative_passages.text: Nested summations and their relation to binomial coefficients
negative_passages.docid: 143573
negative_passages.text: Problem (1.8) A Path to Combinatorics Solution Verification.