Question

Use combinatorial proof to show that $\sum\limits_{k=0}^{n}\binom{n}{k}\binom{k}{m}=\binom{n}{m}2^{n-m}$.

score 142 · Answer 1 · 2022-06-20 23:29:29

142

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Use combinatorial proof to show that $\sum\limits_{k=0}^{n}\binom{n}{k}\binom{k}{m}=\binom{n}{m}2^{n-m}$.

Published on 20 Jun 2022 - 23:29

score 102 · Answer 2 · 2022-06-28 13:56:35

102

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Proof $\tbinom{n}{2}=\tbinom{k}{2}+k(n-k)+\tbinom{n-k}{2}$ based on binomial coefficient definition

Published on 28 Jun 2022 - 13:56

score 57 · Answer 3 · 2022-07-01 06:10:30

57

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Sum with binomial coefficient

Published on 01 Jul 2022 - 6:10

score 228 · Answer 4 · 2022-07-03 06:17:00

228

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Evaluate ${\large\sum_{k=1}^n} \sum_{i=1}^k i{k \choose i}{{n - k - 1} \choose k - i - 1}$

Published on 03 Jul 2022 - 6:17

score 175 · Answer 5 · 2022-07-04 17:09:03

175

Views

How do we meaningfully interpret $\sum_{k = 0}^n x^k = \frac{1 - x^{n+ 1}}{1 - x}$ in the context of combinatorics?

Published on 04 Jul 2022 - 17:09

score 96 · Answer 6 · 2022-07-10 22:17:36

96

Views

Combinatorial Identities involving roots of unity

Published on 10 Jul 2022 - 22:17

score 107 · Answer 7 · 2022-07-11 20:09:08

107

Views

Prove using a combinatorial argument that $\frac{(2n)!}{2^n} \in \mathbb{N}$

Published on 11 Jul 2022 - 20:09

score 337 · Answer 8 · 2022-07-12 01:21:37

337

Views

Prove, using a combinatorial argument, that $\binom{m}{n}\cdot\binom{n}{r}= \binom{m}{r}\cdot \binom{m-r}{n-r}$

Published on 12 Jul 2022 - 1:21

score 136 · Answer 9 · 2022-07-18 14:55:29

136

Views

An identity involving the alternating sum of products of binomial coefficients

Published on 18 Jul 2022 - 14:55

score 61 · Answer 10 · 2022-07-20 17:47:24

61

Views

If $P(n, m)$ is the number of permutations such that $m$ is the first fixed point, prove that $P(n, m + 1) = P(n, m) - P(n - 1, m)$

Published on 20 Jul 2022 - 17:47

score 191 · Answer 11 · 2022-07-28 20:53:47

191

Views

Give combinatorial a proof that $\binom{n}{2}^2=\binom{n}{2}+6\binom{n}{3}+6\binom{n}{4}$

Published on 28 Jul 2022 - 20:53

score 151 · Answer 12 · 2026-02-23 06:38:08

151

Views

Proving the cumulants to moments formula and combinatorics

Published on 23 Feb 2026 - 6:38

score 158 · Answer 13 · 2022-08-04 07:47:13

158

Views

What is another way to prove the least amount of pence, nickels, dimes and quarters that guarantees making any change exactly?

Published on 04 Aug 2022 - 7:47

score 107 · Answer 14 · 2022-08-04 21:00:20

107

Views

prove or disprove that $\sum_{k=0}^n (k-1)^2 p_n(k) = n!$

Published on 04 Aug 2022 - 21:00

score 74 · Answer 15 · 2022-08-06 05:14:41

74

Views

Proving $\sum^n_{k=0} k^2 {n\choose k} = (n+n^2)2^{n-2}$

Published on 06 Aug 2022 - 5:14

List Question

Use combinatorial proof to show that $\sum\limits_{k=0}^{n}\binom{n}{k}\binom{k}{m}=\binom{n}{m}2^{n-m}$.

Proof $\tbinom{n}{2}=\tbinom{k}{2}+k(n-k)+\tbinom{n-k}{2}$ based on binomial coefficient definition

Sum with binomial coefficient

Evaluate ${\large\sum_{k=1}^n} \sum_{i=1}^k i{k \choose i}{{n - k - 1} \choose k - i - 1}$

How do we meaningfully interpret $\sum_{k = 0}^n x^k = \frac{1 - x^{n+ 1}}{1 - x}$ in the context of combinatorics?

Combinatorial Identities involving roots of unity

Prove using a combinatorial argument that $\frac{(2n)!}{2^n} \in \mathbb{N}$

Prove, using a combinatorial argument, that $\binom{m}{n}\cdot\binom{n}{r}= \binom{m}{r}\cdot \binom{m-r}{n-r}$

An identity involving the alternating sum of products of binomial coefficients

If $P(n, m)$ is the number of permutations such that $m$ is the first fixed point, prove that $P(n, m + 1) = P(n, m) - P(n - 1, m)$

Give combinatorial a proof that $\binom{n}{2}^2=\binom{n}{2}+6\binom{n}{3}+6\binom{n}{4}$

Proving the cumulants to moments formula and combinatorics

What is another way to prove the least amount of pence, nickels, dimes and quarters that guarantees making any change exactly?

prove or disprove that $\sum_{k=0}^n (k-1)^2 p_n(k) = n!$

Proving $\sum^n_{k=0} k^2 {n\choose k} = (n+n^2)2^{n-2}$

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