# Tagged Questions

Coefficients involved in the Binomial Theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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### Summation involving binomial coefficients

I came across the following sum online and have spent awhile trying to compute it: $$\sum_{i=0}^{100} \binom{300}{3i}$$ Based on a pattern I noticed, the answer should be $\frac{2^{300}}{3}$ rounded ...
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### Proof of $\sum_{0 \le k \le t} {t-k \choose r}{k \choose s}={t+1 \choose r+s+1}$?

How do I prove that $$\sum_{0 \le k \le t} {t-k \choose r}{k \choose s}={t+1 \choose r+s+1} \>?$$ I saw this in a book discussing generating functions.
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### Equating coefficients of binomial expansion modulo p

In this answer: http://math.stackexchange.com/a/652909 Ted equates mod $p$ the coefficients of $$\sum_{n=0}^{pa} \binom{pa}{n} x^n$$ and $$\sum_{i=0}^{a} \binom{a}{i} x^{pi}$$ to get that ...
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### prove the formula and then evalute the sum

m,n,r are given non-negative integers, show that $\sum_{k>=-n}$ ${r \choose m+k}$ ${s \choose n+k}$ $=$ ${r+s \choose r-m+n}$ Then evaluate $\sum_{k>=0}k$ ${r \choose k}$ ${s \choose k}$ I ...
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### Complicated sum with binomial coefficients

I know how to prove, that $\frac{1}{2^{n}}\cdot\sum\limits_{k=0}^nC_n^k \cdot \sqrt{1+2^{2n}v^{2k}(1-v)^{2(n-k)}}$ tends to 2 if n tends to infinity for $v\in (0,\, 1),\ v\neq 1/2$. This can be proved ...