query: Show that ${n\choose r}={{n-1}\choose{r-1}}+{{n-1}\choose {r}}$.
positive_passages.docid: 424
positive_passages.text: Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$
negative_passages.docid: 24588
negative_passages.text: Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
negative_passages.docid: 89576
negative_passages.text: Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$
negative_passages.docid: 96961
negative_passages.text: A proof for the identity $\sum_{n=r}^{\infty} {n \choose r}^{-1} = \frac{r}{r-1}$
negative_passages.docid: 925
negative_passages.text: Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 92784
negative_passages.text: Intuitive (combinatorial) proof of $2^n=\sum_{k=0}^n {n\choose k}$
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 128093
negative_passages.text: Regarding the values that n and r can take in $\binom{n}{r}$
negative_passages.docid: 54442
negative_passages.text: Sum of $k {n \choose k}$ is $n2^{n-1}$
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: 94784
negative_passages.text: $\sum_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$ using Counting argument
negative_passages.docid: 50525
negative_passages.text: Prove that $(n, n, n-1, n-1, \ldots , 4, 4, 3, 3, 2, 2, 1, 1)$ is a graphic!
negative_passages.docid: 69943
negative_passages.text: why is ${n+1\choose k} = {n\choose k} + {n\choose k-1}$?
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: 134989
negative_passages.text: Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number
negative_passages.docid: 30472
negative_passages.text: Evaluate $\sum_{k=0}^{n} {n \choose k}{m \choose k}$ for a given $n$ and $m$.