Question

Proving $\sum\limits_{k=0}^n \binom{n}{k} = 2^n$ by counting in two ways

score 89 · Answer 1 · 2019-09-09 18:59:04

89

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Proving $\sum\limits_{k=0}^n \binom{n}{k} = 2^n$ by counting in two ways

Published on 09 Sep 2019 - 18:59

score 99 · Answer 2 · 2026-03-26 06:17:46

99

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using sets in combinatorial identity proof

Published on 26 Mar 2026 - 6:17

score 155 · Answer 3 · 2019-09-12 10:13:15

155

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Expression of $x^n+\frac1{x^n}$ by $x+\frac1{x}$ where $n$ is a positive odd number.

Published on 12 Sep 2019 - 10:13

score 66 · Answer 4 · 2019-09-16 02:27:03

66

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Why $\sum_{k=0}^m\sum_{r=k}^{k+n} = \sum_{r=0}^{m+n}\sum_{k=0}^r$?

Published on 16 Sep 2019 - 2:27

score 189 · Answer 5 · 2019-09-19 19:11:39

189

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Finding the Taylor Series Expansion using Binomial Series, then obtaining a subsequent Expansion.

Published on 19 Sep 2019 - 19:11

score 111 · Answer 6 · 2019-09-25 03:19:19

111

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Simplify the following expression $\sum_{j=0}^n 2^n \binom{1/2}{n-j}\binom{j-1/2}{j}$

Published on 25 Sep 2019 - 3:19

score 867 · Answer 7 · 2019-09-26 15:37:58

867

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Prove $\forall n\ge0,43\mid 6^{n+2}+7^{2n+1}$ in three ways

Published on 26 Sep 2019 - 15:37

score 104 · Answer 8 · 2019-09-27 02:30:33

104

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How to prove $\sum_{k=0}^n k^2\cdot{n\choose k} = n\cdot(n+1)\cdot2^{n-2}$ using binomial theorem

Published on 27 Sep 2019 - 2:30

score 38 · Answer 9 · 2019-09-27 04:15:53

38

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How to prove $[x^k] \cdot (1+x)^{a+b} = \sum_{j=0}^k{a\choose j}\cdot{b\choose {k-j}}$

Published on 27 Sep 2019 - 4:15

score 173 · Answer 10 · 2026-03-26 06:32:37

To prove for multichoose $\big(\!{{a+b}\choose k}\!\big)= \sum_{j=0}^k \big(\!{a\choose j}\!\big) \cdot \big(\!{b\choose {k-j}}\!\big)$

score 133 · Answer 11 · 2019-09-29 00:15:50

133

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Understanding binomial expansion

Published on 29 Sep 2019 - 0:15

score 59 · Answer 12 · 2014-03-12 00:59:51

59

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Binomial distribution buy candies randomly

Published on 12 Mar 2014 - 0:59

score 412 · Answer 13 · 2014-03-12 01:09:54

412

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Need some help proving Pascal's Triangle properties

Published on 12 Mar 2014 - 1:09

score 166 · Answer 14 · 2014-03-12 10:50:41

166

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How are lopsided binomials (eg $\binom{n}{n+1})?$ defined?

Published on 12 Mar 2014 - 10:50

score 540 · Answer 15 · 2014-03-12 19:08:17

540

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Asymptotic behavior of $\sum_{k=1}^n \binom{n}{k} \left(\frac{ck}{n}\right)^k$

Published on 12 Mar 2014 - 19:08

List Question

Proving $\sum\limits_{k=0}^n \binom{n}{k} = 2^n$ by counting in two ways

using sets in combinatorial identity proof

Expression of $x^n+\frac1{x^n}$ by $x+\frac1{x}$ where $n$ is a positive odd number.

Why $\sum_{k=0}^m\sum_{r=k}^{k+n} = \sum_{r=0}^{m+n}\sum_{k=0}^r$?

Finding the Taylor Series Expansion using Binomial Series, then obtaining a subsequent Expansion.

Simplify the following expression $\sum_{j=0}^n 2^n \binom{1/2}{n-j}\binom{j-1/2}{j}$

Prove $\forall n\ge0,43\mid 6^{n+2}+7^{2n+1}$ in three ways

How to prove $\sum_{k=0}^n k^2\cdot{n\choose k} = n\cdot(n+1)\cdot2^{n-2}$ using binomial theorem

How to prove $[x^k] \cdot (1+x)^{a+b} = \sum_{j=0}^k{a\choose j}\cdot{b\choose {k-j}}$

To prove for multichoose $\big(\!{{a+b}\choose k}\!\big)= \sum_{j=0}^k \big(\!{a\choose j}\!\big) \cdot \big(\!{b\choose {k-j}}\!\big)$

Understanding binomial expansion

Binomial distribution buy candies randomly

Need some help proving Pascal's Triangle properties

How are lopsided binomials (eg $\binom{n}{n+1})?$ defined?

Asymptotic behavior of $\sum_{k=1}^n \binom{n}{k} \left(\frac{ck}{n}\right)^k$

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