query: Using inclusion-exclusion prove that $\sum_{k=0}^{m}(-1)^{k}\binom{m}{k}\binom{n-k}{r} = \binom{n-m}{r-m}$
positive_passages.docid: 28796
positive_passages.text: Use the Inclusion-Exclusion Principle to show that for any non-negative integers $m, r, n$ such that $m≤r≤n$...
negative_passages.docid: 98082
negative_passages.text: Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$
negative_passages.docid: 136580
negative_passages.text: Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$
negative_passages.docid: 19571
negative_passages.text: Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
negative_passages.docid: 58428
negative_passages.text: Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$
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: 49066
negative_passages.text: Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
negative_passages.docid: 57418
negative_passages.text: How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$
negative_passages.docid: 25004
negative_passages.text: Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically
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: 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: 1139
negative_passages.text: Combinatorial proof of $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^k \binom{n-1-j}{k-j}2^j$
negative_passages.docid: 1158
negative_passages.text: How to prove this binomial identity $\sum_{r=0}^n {r {n \choose r}} = n2^{n-1}$?
negative_passages.docid: 94580
negative_passages.text: Prove that, $\binom{3n}{r}=\binom{3n-1}{r} + \binom{3n-1}{r-1}$
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: 35342
negative_passages.text: Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
negative_passages.docid: 72026
negative_passages.text: Induction proof of $\sum_{k=1}^{n} \binom n k = 2^n -1 $