yspm's blog

By yspm, history, 5 weeks ago, In English

happyguy656 just became a Grandmaster and he wants you to help him to prove the below equation:

$$$\sum_{i=1}^{n} i\binom {n-1}{n-i}\begin{Bmatrix} {n-i}\\{k-1}\end{Bmatrix}=\begin{Bmatrix}n \\ k\end{Bmatrix}+(n-1)\begin{Bmatrix}{n-1}\\ k\end{Bmatrix}$$$

Let $$$\binom nm$$$ be $$$0$$$ when $$$n<m$$$ and n,k are given integers

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By yspm, history, 7 weeks ago, In English

I found the below equation when studying probabilities:

$$$n\binom{n-1}{k-1}\sum\limits_{i=0}^{n-k}\binom{n-k}i (-1)^i\frac{1}{k+i+1}=\frac{k}{n+1}$$$

(Here $$$n,k$$$ are positive integers and $$$n\ge k$$$)

but failed to prove it.

Can anyone offer precious help to me. Thanks

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