Question

Prove ${M}\choose{N_1}$ ${M}\choose{N_2}$ $\leq$ ${2M}\choose{N_1+N_2}$ for $M, N_1, N_2 \in \mathbb{N},\,M\geq N_1,N_2$.

score 237 · Answer 1 · 2020-12-01 15:03:08

237

Views

Prove ${M}\choose{N_1}$ ${M}\choose{N_2}$ $\leq$ ${2M}\choose{N_1+N_2}$ for $M, N_1, N_2 \in \mathbb{N},\,M\geq N_1,N_2$.

Published on 01 Dec 2020 - 15:03

score 94 · Answer 2 · 2020-12-04 13:39:33

94

Views

Is $\sum_{p=0}^{k}(-1)^p\binom{k}{p}p^k=\sum_{p=0}^{k}(-1)^p\binom{k}{p}(p+a)^k, a \in \mathbb{R}$ true? If so, how?

Published on 04 Dec 2020 - 13:39

score 334 · Answer 3 · 2020-12-04 21:36:39

334

Views

N distinct elements, is a permutation or all possible subsets larger?

Published on 04 Dec 2020 - 21:36

score 103 · Answer 4 · 2026-03-28 21:57:10

103

Views

Clustering of vertices in an n dimensional cube

Published on 28 Mar 2026 - 21:57

score 105 · Answer 5 · 2020-12-06 05:18:21

105

Views

A PigeonHole principle (combinatorics)problem using Fibonacci numbers

Published on 06 Dec 2020 - 5:18

score 84 · Answer 6 · 2020-12-06 11:31:29

84

Views

How to divide students into different groups?

Published on 06 Dec 2020 - 11:31

score 142 · Answer 7 · 2020-12-09 21:23:00

142

Views

Combinatorial proof of $ \sum\limits_{k=0}^n(-1)^k\frac{2^{n-k}\binom{n}{k}}{(m+k+1)\binom{m+k}{m}}=\sum\limits_{k=0}^n\frac{\binom{n}{k}}{m+k+1}$

Published on 09 Dec 2020 - 21:23

score 60 · Answer 8 · 2020-12-09 23:05:22

60

Views

Apply the principle of mathematical induction or its variants to show that for all positive integers $m$ and $n$ such that $0\le m<n$:

Published on 09 Dec 2020 - 23:05

score 46 · Answer 9 · 2020-12-10 11:53:20

46

Views

Closed-form expression for a beta-type sum

Published on 10 Dec 2020 - 11:53

score 196 · Answer 10 · 2020-12-10 14:36:05

196

Views

Combinatorial proof of $\sum_{i=0}^m{\binom{x+y+i}{i} \binom{y}{a-i} \binom{x}{b-i} }=\binom{x+a}{b} \binom{y+b}{a}$

Published on 10 Dec 2020 - 14:36

score 244 · Answer 11 · 2026-03-29 18:32:36

244

Views

Combinatorial proof of a nice expression for Stirling numbers of second kind

Published on 29 Mar 2026 - 18:32

score 437 · Answer 12 · 2020-12-12 19:47:37

437

Views

Coefficient of $\frac{1}{1-x-x^2-..-x^{d-1}}$ and its asymptotic

Published on 12 Dec 2020 - 19:47

score 57 · Answer 13 · 2020-12-13 14:38:48

57

Views

Prove $\sum\limits_{k=1}^n \binom{n}{k}k^2=2^{n-2}n(n+1)$ by combinatorial argument.

Published on 13 Dec 2020 - 14:38

score 301 · Answer 14 · 2020-12-16 06:30:47

301

Views

Hint for Spivak's Calculus Problem 2-4-a

Published on 16 Dec 2020 - 6:30

score 77 · Answer 15 · 2020-12-16 23:02:44

77

Views

Prove combinatorically - $\sum_{k=0}^{n} 2^{k}\binom{n}{k} = 3^{n}$

Published on 16 Dec 2020 - 23:02

List Question

Prove ${M}\choose{N_1}$ ${M}\choose{N_2}$ $\leq$ ${2M}\choose{N_1+N_2}$ for $M, N_1, N_2 \in \mathbb{N},\,M\geq N_1,N_2$.

Is $\sum_{p=0}^{k}(-1)^p\binom{k}{p}p^k=\sum_{p=0}^{k}(-1)^p\binom{k}{p}(p+a)^k, a \in \mathbb{R}$ true? If so, how?

N distinct elements, is a permutation or all possible subsets larger?

Clustering of vertices in an n dimensional cube

A PigeonHole principle (combinatorics)problem using Fibonacci numbers

How to divide students into different groups?

Combinatorial proof of $ \sum\limits_{k=0}^n(-1)^k\frac{2^{n-k}\binom{n}{k}}{(m+k+1)\binom{m+k}{m}}=\sum\limits_{k=0}^n\frac{\binom{n}{k}}{m+k+1}$

Apply the principle of mathematical induction or its variants to show that for all positive integers $m$ and $n$ such that $0\le m<n$:

Closed-form expression for a beta-type sum

Combinatorial proof of $\sum_{i=0}^m{\binom{x+y+i}{i} \binom{y}{a-i} \binom{x}{b-i} }=\binom{x+a}{b} \binom{y+b}{a}$

Combinatorial proof of a nice expression for Stirling numbers of second kind

Coefficient of $\frac{1}{1-x-x^2-..-x^{d-1}}$ and its asymptotic

Prove $\sum\limits_{k=1}^n \binom{n}{k}k^2=2^{n-2}n(n+1)$ by combinatorial argument.

Hint for Spivak's Calculus Problem 2-4-a

Prove combinatorically - $\sum_{k=0}^{n} 2^{k}\binom{n}{k} = 3^{n}$

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