query: Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
positive_passages.docid: 46132
positive_passages.text: Combinatorial interpretation for the identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$?
negative_passages.docid: 115698
negative_passages.text: Combinatorics Identity about Catalan numbers: $\sum_{k=0}^n \frac{1}{k+1}\binom{2k}k \binom{2n-2k}{n-k}=\binom{2n+1}n$
negative_passages.docid: 145524
negative_passages.text: Proving this identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$ using lattice paths
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: 96945
negative_passages.text: General formula of $\sum_{n\ge1}{\binom{2n}{n}^{-k}}$.
negative_passages.docid: 60860
negative_passages.text: Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
negative_passages.docid: 89552
negative_passages.text: proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
negative_passages.docid: 39199
negative_passages.text: closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$
negative_passages.docid: 118024
negative_passages.text: Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$
negative_passages.docid: 345
negative_passages.text: Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.
negative_passages.docid: 41354
negative_passages.text: Binomial-coefficients if, k, m, n natural numbers and k \leq n the result of
negative_passages.docid: 30338
negative_passages.text: How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$
negative_passages.docid: 145048
negative_passages.text: Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
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: 126888
negative_passages.text: How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?
negative_passages.docid: 64378
negative_passages.text: Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$
negative_passages.docid: 18046
negative_passages.text: Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$