Question

How to prove this combinatorially $\binom{n}{k}+\binom{n+1}{k}+\binom{n+2}{k}+\cdots+\binom{n+m}{k} = \binom{n+m+1}{k+1}-\binom{n}{k+1}$?

score 113 · Answer 1 · 2020-07-03 12:54:32

113

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How to prove this combinatorially $\binom{n}{k}+\binom{n+1}{k}+\binom{n+2}{k}+\cdots+\binom{n+m}{k} = \binom{n+m+1}{k+1}-\binom{n}{k+1}$?

Published on 03 Jul 2020 - 12:54

score 85 · Answer 2 · 2020-07-04 10:55:32

85

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Determine the number of ordered triples

Published on 04 Jul 2020 - 10:55

score 53 · Answer 3 · 2026-03-25 23:51:02

53

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Trinomial equation - combinatorial explanation

Published on 25 Mar 2026 - 23:51

score 138 · Answer 4 · 2020-07-05 10:40:50

138

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Use the Pascal Identity to prove given equality.

Published on 05 Jul 2020 - 10:40

score 53 · Answer 5 · 2020-07-06 14:44:38

53

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If $\pi$ is a permutation, how many permutation $\sigma$ can be reached from $\pi$ exchanging $r$ indices?

Published on 06 Jul 2020 - 14:44

score 65 · Answer 6 · 2020-07-10 22:36:10

65

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Counting argument to compute the difference $\binom{n}{k} - \binom{\frac{n}{m}}{\frac{k}{m}}^m$

Published on 10 Jul 2020 - 22:36

score 411 · Answer 7 · 2020-07-17 18:28:01

411

Views

Is there a story proof behind the combinatorial identity $(n-2k)\binom{n}{k} = n\left[ \binom{n-1}{k} - \binom{n-1}{k-1} \right]$?

Published on 17 Jul 2020 - 18:28

score 466 · Answer 8 · 2020-07-17 22:12:02

466

Views

Is there an explicit construction of this bijection?

Published on 17 Jul 2020 - 22:12

score 265 · Answer 9 · 2026-03-25 20:18:25

265

Views

Proving $\binom{2n}{n} \geq \frac{2^{2n}}{n+1}$ combinatorially

Published on 25 Mar 2026 - 20:18

score 229 · Answer 10 · 2020-07-22 17:25:52

229

Views

Proving $\log\left(\frac{4^n}{\sqrt{2n+1}{2n\choose n+m}}\right)\geq \frac{m^2}{n}$

Published on 22 Jul 2020 - 17:25

score 98 · Answer 11 · 2020-07-26 22:45:59

98

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Help with proof that $ \sum_{i=0}^k{k \choose i}{N-k \choose n-i} = {N \choose n} $

Published on 26 Jul 2020 - 22:45

score 82 · Answer 12 · 2020-07-27 15:07:38

82

Views

Recursion with combinatoric argument

Published on 27 Jul 2020 - 15:07

score 820 · Answer 13 · 2020-07-29 08:30:41

820

Views

Show by counting two ways that $\sum_{i=1}^{n}i(n-i)=\sum_{i=1}^{n}{i\choose 2}={n+1 \choose 3}$?

Published on 29 Jul 2020 - 8:30

score 88 · Answer 14 · 2020-07-30 02:47:50

88

Views

Prove $\sum_{b=0}^{n-1}b\binom{n+1-b}{2} = \binom{n+2}{4}$

Published on 30 Jul 2020 - 2:47

score 143 · Answer 15 · 2016-07-07 15:00:37

143

Views

Combinatorial analysis

Published on 07 Jul 2016 - 15:00

List Question

How to prove this combinatorially $\binom{n}{k}+\binom{n+1}{k}+\binom{n+2}{k}+\cdots+\binom{n+m}{k} = \binom{n+m+1}{k+1}-\binom{n}{k+1}$?

Determine the number of ordered triples

Trinomial equation - combinatorial explanation

Use the Pascal Identity to prove given equality.

If $\pi$ is a permutation, how many permutation $\sigma$ can be reached from $\pi$ exchanging $r$ indices?

Counting argument to compute the difference $\binom{n}{k} - \binom{\frac{n}{m}}{\frac{k}{m}}^m$

Is there a story proof behind the combinatorial identity $(n-2k)\binom{n}{k} = n\left[ \binom{n-1}{k} - \binom{n-1}{k-1} \right]$?

Is there an explicit construction of this bijection?

Proving $\binom{2n}{n} \geq \frac{2^{2n}}{n+1}$ combinatorially

Proving $\log\left(\frac{4^n}{\sqrt{2n+1}{2n\choose n+m}}\right)\geq \frac{m^2}{n}$

Help with proof that $ \sum_{i=0}^k{k \choose i}{N-k \choose n-i} = {N \choose n} $

Recursion with combinatoric argument

Show by counting two ways that $\sum_{i=1}^{n}i(n-i)=\sum_{i=1}^{n}{i\choose 2}={n+1 \choose 3}$?

Prove $\sum_{b=0}^{n-1}b\binom{n+1-b}{2} = \binom{n+2}{4}$

Combinatorial analysis

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