query: Using Binomial Theorem to find the value of $ ^nC_0^2 - ^nC_1^2 + ^nC_2^2 - ^nC_3^2... + (-1)^n . ^nC_n ^2 $
positive_passages.docid: 31970
positive_passages.text: Alternating sum of squares of binomial coefficients
negative_passages.docid: 19581
negative_passages.text: $\sum_{k=0}^n (-1)^k \binom{n}{k}^2$ and $\sum_{k=0}^n k \binom{n}{k}^2$
negative_passages.docid: 128913
negative_passages.text: A proof of $\sum_{i=1}^{n}(-1)^{i-1}\binom{n}{i}\frac{1}{i}=\frac{1}{1}+\frac{1}{2}+....+\frac{1}{n}$
negative_passages.docid: 43066
negative_passages.text: Finding value of $S-T$ in $2$ binomial sum.
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: 74081
negative_passages.text: Find coefficient of $x^{n}$ in $\frac{1}{(1-x)^3(1+x)^2}$
negative_passages.docid: 52557
negative_passages.text: Derivation of binomial coefficient in binomial theorem.
negative_passages.docid: 39199
negative_passages.text: closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$
negative_passages.docid: 35342
negative_passages.text: Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
negative_passages.docid: 112827
negative_passages.text: Find the sum of the binomial coefficients
negative_passages.docid: 113943
negative_passages.text: Prove this equality by using Newton's Binomial Theorem
negative_passages.docid: 44111
negative_passages.text: Find the value of $\sum_{k=1}^{n}k\binom{n}{k}$?
negative_passages.docid: 118372
negative_passages.text: Combination proof for $n(n+1)2^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}$
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: 91458
negative_passages.text: Sum of the series $\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{2}-\binom{n-3}{3}+..........$
negative_passages.docid: 143927
negative_passages.text: Binomial coefficients that are powers of 2
negative_passages.docid: 112374
negative_passages.text: Evaluate $ \binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\cdots+\binom{n}{2k}+\cdots$