query: Evaluate $\binom{n}{1} + 2\binom{n}{2} + 3 \binom{n}{3}+...+ n \binom{n}{n}$
positive_passages.docid: 155769
positive_passages.text: Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
negative_passages.docid: 14399
negative_passages.text: Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$
negative_passages.docid: 74294
negative_passages.text: How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$?
negative_passages.docid: 72639
negative_passages.text: Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
negative_passages.docid: 122891
negative_passages.text: integer as sum of three binomials
negative_passages.docid: 119107
negative_passages.text: Is it true that $\sum_{k=3}^{n}\binom{n}{k}\binom{k}{3} = \binom{n}{3}\binom{n-3}{k-3}$?
negative_passages.docid: 64803
negative_passages.text: Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
negative_passages.docid: 94580
negative_passages.text: Prove that, $\binom{3n}{r}=\binom{3n-1}{r} + \binom{3n-1}{r-1}$
negative_passages.docid: 13236
negative_passages.text: Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$
negative_passages.docid: 15985
negative_passages.text: simplify $(a_1 + a_2 +a_3+... +a_n)^m$
negative_passages.docid: 58247
negative_passages.text: Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
negative_passages.docid: 10917
negative_passages.text: Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
negative_passages.docid: 56566
negative_passages.text: If ${n \choose 3} + {n+3-1 \choose 3} = (n)_3$, compute $n$.
negative_passages.docid: 115433
negative_passages.text: Find the summation of given expression: $\sum_{k=1}^n k^3\binom nk$
negative_passages.docid: 44111
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 $\binom{n}{0}\binom{n}{1}+\binom{n}{1}\binom{n}{2}+...+\binom{n}{n-1}\binom{n}{n}$
negative_passages.docid: 118024
negative_passages.text: Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$