query: Find $n$ such that $ \binom{2n+1}{n+2} = \binom{2n+1}{5} $
positive_passages.docid: 156873
positive_passages.text: Solve the equation $\binom{2n+1}{n+2} = \binom{2n+1}{5} $
negative_passages.docid: 31196
negative_passages.text: Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$
negative_passages.docid: 139426
negative_passages.text: Proving $ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $ without induction
negative_passages.docid: 120074
negative_passages.text: Prove by Combinatorial Argument that $\binom{n}{k}= \frac{n}{k} \binom{n-1}{k-1}$
negative_passages.docid: 44790
negative_passages.text: Find $n$ for which $\frac{n(n+1)}{2}$ is perfect square
negative_passages.docid: 111098
negative_passages.text: Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
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: 125220
negative_passages.text: Solving binomial theorem via induction
negative_passages.docid: 133420
negative_passages.text: Computing the binomial series: $\binom{n}{2}\cdot2^2+\binom{n}{4}\cdot2^4+\binom{n}{6}\cdot2^6+\cdots+ \binom{n}{n}\cdot2^n$
negative_passages.docid: 143501
negative_passages.text: Find all $n\in\mathbb N^+$ such that the sum of the digits of $5^n$ is equal to $2^n$.
negative_passages.docid: 61871
negative_passages.text: Formula for the $n$th term of $1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5, ...$
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: 143365
negative_passages.text: Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?
negative_passages.docid: 114277
negative_passages.text: Proving in Discrete Structures Problem that $2^N = \binom{N}{0} + \binom{N}{1} + \binom{N}{2} + \dots + \binom{N}{N}$
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: 55481
negative_passages.text: Prove the identity $\binom{2n+1}{0} + \binom{2n+1}{1} + \cdots + \binom{2n+1}{n} = 4^n$
negative_passages.docid: 92704
negative_passages.text: Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$