query: What is the combinatorial interpretation for this expression$ \sum_{i=0}^{n} (-1)^{i} \binom{n}{i} = 0$
positive_passages.docid: 116461
positive_passages.text: Prove the identity $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=0$ using a combinatorial interpretation of the positive and negative terms
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 36388
negative_passages.text: How to derive this Combinatorial identity
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: 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: 116491
negative_passages.text: Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$
negative_passages.docid: 139842
negative_passages.text: Seeking a combinatorial proof $\sum\limits _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$
negative_passages.docid: 87589
negative_passages.text: Evaluate $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+...+\binom{n}{2k}$
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: 122236
negative_passages.text: How to compute the series: $\sum_{n=0}^{\infty} (-1)^{n-1}\binom{1/2}{n}$
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: 146762
negative_passages.text: Closed form for $\sum_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{(k+1)^2}$
negative_passages.docid: 40784
negative_passages.text: $\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
negative_passages.docid: 146533
negative_passages.text: Having problems proving $ \sum_{r=0}^{n}(r+1) \binom{n}{r} = (n+2)2^{n-1} $
negative_passages.docid: 136580
negative_passages.text: Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$
negative_passages.docid: 148949
negative_passages.text: Find the value of $\sum_{k=1}^{n} k \binom {n} {k}$
negative_passages.docid: 75367
negative_passages.text: Find a simple formula for $\binom{n}{0}\binom{n}{1}+\binom{n}{1}\binom{n}{2}+...+\binom{n}{n-1}\binom{n}{n}$