query: Combinatorial proof for $2^n=1+\sum_{k=0}^{n-1}2^k$
positive_passages.docid: 108318
positive_passages.text: counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
negative_passages.docid: 59252
negative_passages.text: Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
negative_passages.docid: 84010
negative_passages.text: Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
negative_passages.docid: 58247
negative_passages.text: Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
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: 74887
negative_passages.text: Combinatorial proof of the identity ${n+1\choose k+1}={k\choose k}+{k+1\choose k}+...+{n-1\choose k}+{n\choose k}$
negative_passages.docid: 111098
negative_passages.text: Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
negative_passages.docid: 146200
negative_passages.text: proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$
negative_passages.docid: 65983
negative_passages.text: Combinatorial proof of binomial coefficient summation identity
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: 61483
negative_passages.text: Prove the following identity combinatorially
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: 120976
negative_passages.text: Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
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: 1167
negative_passages.text: How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
negative_passages.docid: 1837
negative_passages.text: Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
negative_passages.docid: 70827
negative_passages.text: Prove that $\sum_{i = n - b}^{a - m} {a - i \choose m}{b + i \choose n} = {a + b + 1 \choose m + n + 1}$