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Computing the binomial sum $\sum_{0\le i<j\le n}j\binom ni$

score 691 · Answer 1 · 2026-03-30 06:10:17

691

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Computing the binomial sum $\sum_{0\le i<j\le n}j\binom ni$

Published on 30 Mar 2026 - 6:10

score 52 · Answer 2 · 2026-03-25 10:53:38

52

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Matrix with characteristic equation equal to the rows of pascal triangle

Published on 25 Mar 2026 - 10:53

score 107 · Answer 3 · 2026-03-28 00:49:12

107

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Why is $\lim_{n\to\infty}\binom{n}{i} \frac{1}{n^i} \left (1-\frac{1}{n}\right)^{n-i}=\frac{1}{i!} e^{-1}$?

Published on 28 Mar 2026 - 0:49

score 91 · Answer 4 · 2020-04-06 11:32:35

91

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Determining the PMF of an internet service provider using 50 modems to serve 1000 customers.

Published on 06 Apr 2020 - 11:32

score 28 · Answer 5 · 2020-04-07 09:32:28

28

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Binomial coefficient: Calculate and save in the simplest possible form:

Published on 07 Apr 2020 - 9:32

score 155 · Answer 6 · 2020-04-07 20:26:34

155

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Solving recurrence relation in 2 variables plus constant

Published on 07 Apr 2020 - 20:26

score 141 · Answer 7 · 2026-03-30 07:37:42

141

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Analytically: How does Pascal's triangle cook Sierpinski's triangle?

Published on 30 Mar 2026 - 7:37

score 86 · Answer 8 · 2026-04-01 17:44:21

86

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Proving an inequality involving real powers

Published on 01 Apr 2026 - 17:44

score 75 · Answer 9 · 2026-03-28 10:54:10

75

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Hypergeometric representation degenerate cases

Published on 28 Mar 2026 - 10:54

score 149 · Answer 10 · 2026-03-28 02:20:59

149

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Binomial expansion and combinatorics

Published on 28 Mar 2026 - 2:20

score 169 · Answer 11 · 2026-03-30 07:35:57

169

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A 'Difficult' Binomial Sum - Variable Base and Exponent

Published on 30 Mar 2026 - 7:35

score 170 · Answer 12 · 2026-03-30 07:41:29

170

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Find $r$ which maximizes $\binom{20}r\binom{20}0+\binom{20}{r-1}\binom{20}1+\binom{20}{r-2}\binom{20}2+\cdots+\binom{20}0\binom{20}r$

Published on 30 Mar 2026 - 7:41

score 59 · Answer 13 · 2020-04-09 16:37:14

59

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Combinatorics summation problem.

Published on 09 Apr 2020 - 16:37

score 224 · Answer 14 · 2026-03-26 08:03:22

224

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Expansion of $(a+b+c+d+e+....)^n$, but with all coefficients equal to 1.

Published on 26 Mar 2026 - 8:03

score 326 · Answer 15 · 2020-04-10 17:21:16

326

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Proving $\sum_{k=0}^n\sum_{l=0}^k{n \choose k}^2{k \choose l}{n \choose l}{2n-l \choose n}=\sum_{k=0}^n {n \choose k}^2{n+k \choose k}^2$

Published on 10 Apr 2020 - 17:21

List Question

Computing the binomial sum $\sum_{0\le i<j\le n}j\binom ni$

Matrix with characteristic equation equal to the rows of pascal triangle

Why is $\lim_{n\to\infty}\binom{n}{i} \frac{1}{n^i} \left (1-\frac{1}{n}\right)^{n-i}=\frac{1}{i!} e^{-1}$?

Determining the PMF of an internet service provider using 50 modems to serve 1000 customers.

Binomial coefficient: Calculate and save in the simplest possible form:

Solving recurrence relation in 2 variables plus constant

Analytically: How does Pascal's triangle cook Sierpinski's triangle?

Proving an inequality involving real powers

Hypergeometric representation degenerate cases

Binomial expansion and combinatorics

A 'Difficult' Binomial Sum - Variable Base and Exponent

Find $r$ which maximizes $\binom{20}r\binom{20}0+\binom{20}{r-1}\binom{20}1+\binom{20}{r-2}\binom{20}2+\cdots+\binom{20}0\binom{20}r$

Combinatorics summation problem.

Expansion of $(a+b+c+d+e+....)^n$, but with all coefficients equal to 1.

Proving $\sum_{k=0}^n\sum_{l=0}^k{n \choose k}^2{k \choose l}{n \choose l}{2n-l \choose n}=\sum_{k=0}^n {n \choose k}^2{n+k \choose k}^2$

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