query: Combinatorial proof of $ k{n \choose k} = n {n-1\choose k-1} $
positive_passages.docid: 70282
positive_passages.text: Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$
negative_passages.docid: 56828
negative_passages.text: A combinatorial proof that the alternating sum of binomial coefficients is zero
negative_passages.docid: 92784
negative_passages.text: Intuitive (combinatorial) proof of $2^n=\sum_{k=0}^n {n\choose k}$
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: 6366
negative_passages.text: Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
negative_passages.docid: 20317
negative_passages.text: Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
negative_passages.docid: 54442
negative_passages.text: Sum of $k {n \choose k}$ is $n2^{n-1}$
negative_passages.docid: 9522
negative_passages.text: Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
negative_passages.docid: 106008
negative_passages.text: Combinatorial Identity $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$
negative_passages.docid: 87320
negative_passages.text: Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially
negative_passages.docid: 116356
negative_passages.text: Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$
negative_passages.docid: 142534
negative_passages.text: Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
negative_passages.docid: 75910
negative_passages.text: Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
negative_passages.docid: 112573
negative_passages.text: Combinatorial proof for Stirling number of 1st kind's generating function
negative_passages.docid: 82872
negative_passages.text: Combinatorial proof involving partitions and generating functions
negative_passages.docid: 107397
negative_passages.text: Algebraic proof for the identity $\sum_{k=0}^m(-1)^{m-k}{{n \choose {k}}}= {{n-1}\choose m}$
negative_passages.docid: 1125
negative_passages.text: Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$