Question

Can you not apply $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$ to the series $\sum^{n+1}_{k=0}\binom{n+1}{k}$

score 86 · Answer 1 · 2026-04-16 05:42:10

86

Views

Can you not apply $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$ to the series $\sum^{n+1}_{k=0}\binom{n+1}{k}$

Published on 16 Apr 2026 - 5:42

score 121 · Answer 2 · 2026-04-16 18:52:54

121

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Prove that $\sum_{i=0}^{50} {50 \choose i}^2 = {100 \choose 50}$

Published on 16 Apr 2026 - 18:52

score 95 · Answer 3 · 2026-04-16 06:57:02

95

Views

A nice proof of the identity $\sum_{i=0}^{l+1}(-1)^i\binom{n+i}{l}\binom{l+1}{i}=0$

Published on 16 Apr 2026 - 6:57

score 669 · Answer 4 · 2026-04-11 17:36:09

669

Views

An Alternate Definition for n!

Published on 11 Apr 2026 - 17:36

score 71 · Answer 5 · 2026-04-16 10:45:18

71

Views

Find $\lim\limits_{n\rightarrow\infty}\sqrt[n]{\sum_{k=0}^{n} \frac{(-1)^k}{k+1}\cdot2^{n-k}\cdot\binom{n }{k}}$

Published on 16 Apr 2026 - 10:45

score 41 · Answer 6 · 2026-04-15 05:56:47

41

Views

Where does binomial formula for the probability that, in $n$ coin flips, the number of heads is divisble by 3 come from?

Published on 15 Apr 2026 - 5:56

score 186 · Answer 7 · 2026-04-13 17:07:40

186

Views

Sum with Binomial Coefficients and Sine; $S=\sum_{k=0}^n \binom{n}{k} \sin(kx)$

Published on 13 Apr 2026 - 17:07

score 82 · Answer 8 · 2026-04-17 13:33:55

82

Views

Binomial coefficients relationship to combinations

Published on 17 Apr 2026 - 13:33

score 51 · Answer 9 · 2026-04-12 08:26:26

51

Views

How to prove that this binomial sum remains positive for $x>1$?

Published on 12 Apr 2026 - 8:26

score 126 · Answer 10 · 2026-04-08 17:31:26

126

Views

prove a fibonomial identity

Published on 08 Apr 2026 - 17:31

score 71 · Answer 11 · 2026-03-26 04:38:12

71

Views

Find the value of $\sum_{d=0}^\infty\binom{d+m-1}{m-1}\frac{x^d}{d!}$.

Published on 26 Mar 2026 - 4:38

score 87 · Answer 12 · 2026-04-17 09:03:47

87

Views

Prove $\sum_{k=0}^s\binom{s}{k}\binom{t}{r+k}^{-1}=\frac{t+1}{t-s+1}\binom{t-s}{r}^{-1}$ for $0\leq r,s,r+s\leq t$

Published on 17 Apr 2026 - 9:03

score 46 · Answer 13 · 2026-04-17 13:25:10

46

Views

Is the answer $5!-2\cdot 3! \cdot {4 \choose 1}$ or $5!-2\cdot 3! \cdot 4$ (multiplication rule in combinatorics)?

Published on 17 Apr 2026 - 13:25

score 79 · Answer 14 · 2026-03-26 06:21:05

79

Views

is there a general combinatorics formula like n choose k but instead we choose k,l,m ect.

Published on 26 Mar 2026 - 6:21

score 81 · Answer 15 · 2026-04-16 12:44:53

81

Views

Show that $\sum\limits_{i=0}^{n}\binom{n}{i}(-1)^i (n-i)^k=0$ for $0\leq k<n$

Published on 16 Apr 2026 - 12:44

List Question

Can you not apply $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$ to the series $\sum^{n+1}_{k=0}\binom{n+1}{k}$

Prove that $\sum_{i=0}^{50} {50 \choose i}^2 = {100 \choose 50}$

A nice proof of the identity $\sum_{i=0}^{l+1}(-1)^i\binom{n+i}{l}\binom{l+1}{i}=0$

An Alternate Definition for n!

Find $\lim\limits_{n\rightarrow\infty}\sqrt[n]{\sum_{k=0}^{n} \frac{(-1)^k}{k+1}\cdot2^{n-k}\cdot\binom{n }{k}}$

Where does binomial formula for the probability that, in $n$ coin flips, the number of heads is divisble by 3 come from?

Sum with Binomial Coefficients and Sine; $S=\sum_{k=0}^n \binom{n}{k} \sin(kx)$

Binomial coefficients relationship to combinations

How to prove that this binomial sum remains positive for $x>1$?

prove a fibonomial identity

Find the value of $\sum_{d=0}^\infty\binom{d+m-1}{m-1}\frac{x^d}{d!}$.

Prove $\sum_{k=0}^s\binom{s}{k}\binom{t}{r+k}^{-1}=\frac{t+1}{t-s+1}\binom{t-s}{r}^{-1}$ for $0\leq r,s,r+s\leq t$

Is the answer $5!-2\cdot 3! \cdot {4 \choose 1}$ or $5!-2\cdot 3! \cdot 4$ (multiplication rule in combinatorics)?

is there a general combinatorics formula like n choose k but instead we choose k,l,m ect.

Show that $\sum\limits_{i=0}^{n}\binom{n}{i}(-1)^i (n-i)^k=0$ for $0\leq k<n$

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