Question

Induction based on combinations and binomial theorem: ${}^nC_0+{}^{n+1}C_1+{}^{n+2}C_2+\dots+{}^{n+p}C_p={}^{n+p+1}C_p$

score 108 · Answer 1 · 2018-08-18 07:43:51

108

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Induction based on combinations and binomial theorem: ${}^nC_0+{}^{n+1}C_1+{}^{n+2}C_2+\dots+{}^{n+p}C_p={}^{n+p+1}C_p$

Published on 18 Aug 2018 - 7:43

score 42 · Answer 2 · 2018-08-18 11:11:14

42

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show that $\sum^\infty_{i=1}p^2(1-p)^{2(i-1)}=\frac{p^2}{1-(1-p)^2}$

Published on 18 Aug 2018 - 11:11

score 690 · Answer 3 · 2018-08-18 13:10:17

690

Views

Evaluate the triple sum $\sum_{i=0}^{n-1}\sum_{j=i}^{n-1}\sum_{k=i}^{j} 1$

Published on 18 Aug 2018 - 13:10

score 72 · Answer 4 · 2018-08-18 22:25:25

72

Views

A finite alternate binomial sum with finite value (modulo 6)

Published on 18 Aug 2018 - 22:25

score 37 · Answer 5 · 2026-03-25 01:34:05

37

Views

I need help simplifying a sum problem that involves a binomial raised to a power

Published on 25 Mar 2026 - 1:34

score 1.6k · Answer 6 · 2018-08-19 17:56:29

1.6k

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Binomial Coefficients involving Prime Powers Minus 1

Published on 19 Aug 2018 - 17:56

score 52 · Answer 7 · 2018-08-20 01:39:04

52

Views

A sum that includes power of binomials: Possible limit?

Published on 20 Aug 2018 - 1:39

score 383 · Answer 8 · 2026-03-25 16:21:16

383

Views

Sum of binomial coefficients in Gould tables

Published on 25 Mar 2026 - 16:21

score 132 · Answer 9 · 2026-03-25 16:21:19

132

Views

Asymptotic approximation of a sum involving log and binomial coefficient

Published on 25 Mar 2026 - 16:21

score 75 · Answer 10 · 2026-03-25 15:54:39

75

Views

What's equal $(x+y)^{\sqrt{2}} $ to and what's its geometric interpretation?

Published on 25 Mar 2026 - 15:54

score 113 · Answer 11 · 2026-03-26 20:41:58

113

Views

Prove $\sum_{r=0}^n \binom{n}{r} \binom{n+r}{r} (-2)^r =(-1)^n\sum_{r=0}^n \binom{n}{r}^2 2^r$

Published on 26 Mar 2026 - 20:41

score 105 · Answer 12 · 2018-08-25 08:29:17

105

Views

Understanding solution of binomial problem

Published on 25 Aug 2018 - 8:29

score 429 · Answer 13 · 2026-03-25 11:13:42

429

Views

Combinatorial proof of Catalan number/central binomial convolution: $2n C_n=\sum_{k=1}^n\binom{2k}kC_{n-k}$

Published on 25 Mar 2026 - 11:13

score 55 · Answer 14 · 2018-08-26 17:14:15

55

Views

Summation of Binomial Distribution 2

Published on 26 Aug 2018 - 17:14

score 201 · Answer 15 · 2026-03-25 17:42:27

201

Views

Proving that $\sum_{n=0}^r (-1)^n \binom{r}{n} (s+r-n-1)!/(s-n)! = 0$ without Taylor expansion

Published on 25 Mar 2026 - 17:42

List Question

Induction based on combinations and binomial theorem: ${}^nC_0+{}^{n+1}C_1+{}^{n+2}C_2+\dots+{}^{n+p}C_p={}^{n+p+1}C_p$

show that $\sum^\infty_{i=1}p^2(1-p)^{2(i-1)}=\frac{p^2}{1-(1-p)^2}$

Evaluate the triple sum $\sum_{i=0}^{n-1}\sum_{j=i}^{n-1}\sum_{k=i}^{j} 1$

A finite alternate binomial sum with finite value (modulo 6)

I need help simplifying a sum problem that involves a binomial raised to a power

Binomial Coefficients involving Prime Powers Minus 1

A sum that includes power of binomials: Possible limit?

Sum of binomial coefficients in Gould tables

Asymptotic approximation of a sum involving log and binomial coefficient

What's equal $(x+y)^{\sqrt{2}} $ to and what's its geometric interpretation?

Prove $\sum_{r=0}^n \binom{n}{r} \binom{n+r}{r} (-2)^r =(-1)^n\sum_{r=0}^n \binom{n}{r}^2 2^r$

Understanding solution of binomial problem

Combinatorial proof of Catalan number/central binomial convolution: $2n C_n=\sum_{k=1}^n\binom{2k}kC_{n-k}$

Summation of Binomial Distribution 2

Proving that $\sum_{n=0}^r (-1)^n \binom{r}{n} (s+r-n-1)!/(s-n)! = 0$ without Taylor expansion

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