query: Coefficients in binomial expansions of $(x+y)^n$
positive_passages.docid: 151985
positive_passages.text: If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
negative_passages.docid: 112827
negative_passages.text: Find the sum of the binomial coefficients
negative_passages.docid: 75723
negative_passages.text: Odd Binomial Coefficients?
negative_passages.docid: 91239
negative_passages.text: Sum of binomial coefficients involving $n,p,q,r$
negative_passages.docid: 94060
negative_passages.text: Sum of square binomial coefficients
negative_passages.docid: 142303
negative_passages.text: Alternating sum of binomial coefficients multiplied by (1/k+1)
negative_passages.docid: 27922
negative_passages.text: prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
negative_passages.docid: 46132
negative_passages.text: Combinatorial interpretation for the identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$?
negative_passages.docid: 71564
negative_passages.text: Combinatorial proof of identity involving central binomial coefficients
negative_passages.docid: 99502
negative_passages.text: Question on the coefficients of $(1+x+x^2+x^3+x^4)^{496}$
negative_passages.docid: 125712
negative_passages.text: Working with binomial coefficient $\sum_{k=0}^n (-1)^k \binom nk=0$
negative_passages.docid: 45672
negative_passages.text: Even/Odd Binomial Coefficients
negative_passages.docid: 65983
negative_passages.text: Combinatorial proof of binomial coefficient summation identity
negative_passages.docid: 143927
negative_passages.text: Binomial coefficients that are powers of 2
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: 31970
negative_passages.text: Alternating sum of squares of binomial coefficients
negative_passages.docid: 109599
negative_passages.text: A finite sum involving the binomial coefficients and the harmonic numbers