Question

How to prove combinatorial identity $\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose s}{s\choose m-s}$?

score 2.7k · Answer 1 · 2014-12-12 08:04:54

2.7k

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How to prove combinatorial identity $\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose s}{s\choose m-s}$?

Published on 12 Dec 2014 - 8:04

score 342 · Answer 2 · 2014-12-12 11:38:02

342

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How to prove this Catalan number identity

Published on 12 Dec 2014 - 11:38

score 273 · Answer 3 · 2026-04-04 04:57:28

273

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Proving that $\Phi_{t}[\mathbb{Y},\mathbb{Z}]=[\Phi_{t}\mathbb{Y},\Phi_{t*}\mathbb{Z}]$

Published on 04 Apr 2026 - 4:57

score 194 · Answer 4 · 2014-12-14 07:51:30

194

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Identity of sum of binomial coefficients

Published on 14 Dec 2014 - 7:51

score 124 · Answer 5 · 2014-12-14 11:31:26

124

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Proving a formula with binomial coefficient

Published on 14 Dec 2014 - 11:31

score 1.3k · Answer 6 · 2026-03-29 06:30:27

1.3k

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Prove that $\binom n2 + \binom {n-1}2$ is always a perfect square

Published on 29 Mar 2026 - 6:30

score 184 · Answer 7 · 2014-12-17 11:25:33

184

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Simplifying $\sum_{j=k}^{n}\binom{j}{k}/(2^{k-1})$

Published on 17 Dec 2014 - 11:25

score 13.5k · Answer 8 · 2014-12-17 17:44:12

13.5k

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Probability that given a 1000 page book with 1000 misprints, a page will have 3 misprints.

Published on 17 Dec 2014 - 17:44

score 195 · Answer 9 · 2014-12-17 17:45:47

195

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Another sum involving binomial coefficients.

Published on 17 Dec 2014 - 17:45

score 61 · Answer 10 · 2014-12-18 01:01:19

61

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What can you conclude about the first moment of a variable given the 3rd moment exists and is finite

Published on 18 Dec 2014 - 1:01

score 3.5k · Answer 11 · 2014-12-18 03:06:20

3.5k

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Is there a closed-form formula for sum of "odd combinations"?

Published on 18 Dec 2014 - 3:06

score 142 · Answer 12 · 2014-12-19 03:36:30

142

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Can this binomial summation be simplified?

Published on 19 Dec 2014 - 3:36

score 256 · Answer 13 · 2026-03-31 15:12:37

256

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How to transform the product to sum?

Published on 31 Mar 2026 - 15:12

score 181 · Answer 14 · 2014-12-21 07:47:44

181

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How find all postive integer number such $(n+k)\nmid \binom{2n}{n}$

Published on 21 Dec 2014 - 7:47

score 81 · Answer 15 · 2014-12-21 11:40:52

81

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If given $\sum_{r=1}^{m-1}\binom r3$, how does the summation evaluate when $n<r$ in $\binom nr$?

Published on 21 Dec 2014 - 11:40

List Question

How to prove combinatorial identity $\sum_{k=0}^s{s\choose k}{m\choose k}{k\choose m-s}={2s\choose s}{s\choose m-s}$?

How to prove this Catalan number identity

Proving that $\Phi_{t}[\mathbb{Y},\mathbb{Z}]=[\Phi_{t}\mathbb{Y},\Phi_{t*}\mathbb{Z}]$

Identity of sum of binomial coefficients

Proving a formula with binomial coefficient

Prove that $\binom n2 + \binom {n-1}2$ is always a perfect square

Simplifying $\sum_{j=k}^{n}\binom{j}{k}/(2^{k-1})$

Probability that given a 1000 page book with 1000 misprints, a page will have 3 misprints.

Another sum involving binomial coefficients.

What can you conclude about the first moment of a variable given the 3rd moment exists and is finite

Is there a closed-form formula for sum of "odd combinations"?

Can this binomial summation be simplified?

How to transform the product to sum?

How find all postive integer number such $(n+k)\nmid \binom{2n}{n}$

If given $\sum_{r=1}^{m-1}\binom r3$, how does the summation evaluate when $n<r$ in $\binom nr$?

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