query: Evaluate ${k \choose 0} + {k \choose 2} + {k \choose 4}+ {k \choose 6} +...+ {k \choose k}$ for even $k$
positive_passages.docid: 17638
positive_passages.text: Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
negative_passages.docid: 38705
negative_passages.text: Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$
negative_passages.docid: 4991
negative_passages.text: Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$
negative_passages.docid: 3786
negative_passages.text: Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$
negative_passages.docid: 70282
negative_passages.text: Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$
negative_passages.docid: 83837
negative_passages.text: Odd/Even Permutations
negative_passages.docid: 89552
negative_passages.text: proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$
negative_passages.docid: 24588
negative_passages.text: Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
negative_passages.docid: 6366
negative_passages.text: Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
negative_passages.docid: 25826
negative_passages.text: Proof of a combinatorial identity: $\sum\limits_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$
negative_passages.docid: 104704
negative_passages.text: Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially
negative_passages.docid: 90519
negative_passages.text: Combinatorial proof of $\sum\limits_{k=0}^n {n \choose k}3^k=4^n$
negative_passages.docid: 124467
negative_passages.text: Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$?
negative_passages.docid: 5054
negative_passages.text: Calculate $\sum\limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$
negative_passages.docid: 68080
negative_passages.text: Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$
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: 19868
negative_passages.text: Evaluate: $ \sum _ {k=1}^{n}{\frac{k}{n}\binom{n}{k}t^k(1-t)^{n-k}}$