query: Prove that $k\binom{n}{k}=n\binom{n-1}{k-1}$
positive_passages.docid: 142534
positive_passages.text: Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
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: 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: 107335
negative_passages.text: Prove equality between binomial coefficients.
negative_passages.docid: 117761
negative_passages.text: When is $\binom{n}{k}$ divisible by $n$?
negative_passages.docid: 118140
negative_passages.text: Prove that $ \frac{n(n-1)(n-k+1)}{k(k-1)\cdots1} = \frac{n!}{(n-1)!k!}$
negative_passages.docid: 25961
negative_passages.text: Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$
negative_passages.docid: 133181
negative_passages.text: Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$
negative_passages.docid: 128913
negative_passages.text: A proof of $\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+....+\frac{1}{n}$
negative_passages.docid: 107071
negative_passages.text: Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem
negative_passages.docid: 11612
negative_passages.text: How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
negative_passages.docid: 5149
negative_passages.text: Show $\binom{a}{b}\binom{b}{c}=\binom{a}{a-c}\binom{a-c}{a-b}$
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: 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: 37569
negative_passages.text: Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
negative_passages.docid: 3786
negative_passages.text: Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$
negative_passages.docid: 959
negative_passages.text: Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.