query: Sum of binomial coefficient with both the upper and lower index being multiple of 4
positive_passages.docid: 108275
positive_passages.text: How do I count the subsets of a set whose number of elements is divisible by 3? 4?
negative_passages.docid: 59252
negative_passages.text: Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
negative_passages.docid: 35381
negative_passages.text: Sum with binomial coefficients: $\sum_{k=1}^m  \frac{1}{k}{m \choose k} $
negative_passages.docid: 87479
negative_passages.text: Closed form for $\sum_{k=0}^{n-p}\binom{n}{k}\binom{n}{p+k}$
negative_passages.docid: 143927
negative_passages.text: Binomial coefficients that are powers of 2
negative_passages.docid: 44111
negative_passages.text: Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?
negative_passages.docid: 626
negative_passages.text: Sum of combinations with varying $n$
negative_passages.docid: 56828
negative_passages.text: A combinatorial proof that the alternating sum of binomial coefficients is zero
negative_passages.docid: 142303
negative_passages.text: Alternating sum of binomial coefficients multiplied by (1/k+1)
negative_passages.docid: 151384
negative_passages.text: Summation of series involving binomial coefficients and polynomial of degree at most n-1
negative_passages.docid: 112374
negative_passages.text: Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$
negative_passages.docid: 83229
negative_passages.text: Nested summations and their relation to binomial coefficients
negative_passages.docid: 18793
negative_passages.text: Partial sum of rows of Pascal's triangle
negative_passages.docid: 57549
negative_passages.text: Finding the sum $\sum_{k=1}^rk^2\binom {n-k}{r-k}$
negative_passages.docid: 144571
negative_passages.text: A conjecture including binomial coefficients
negative_passages.docid: 71797
negative_passages.text: Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
negative_passages.docid: 83400
negative_passages.text: Sum of products of binomial coefficients: $ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j} $