Question

How to prove that $\sum_{j=0}^{k}{(-1)^j\binom{n}{j}}=(-1)^k\binom{n-1}{k}$?

score 44 · Answer 1 · 2026-04-15 07:03:17

44

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How to prove that $\sum_{j=0}^{k}{(-1)^j\binom{n}{j}}=(-1)^k\binom{n-1}{k}$?

Published on 15 Apr 2026 - 7:03

score 41 · Answer 2 · 2026-04-14 07:58:18

41

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Proving a summation series involving binomial coefficients

Published on 14 Apr 2026 - 7:58

score 80 · Answer 3 · 2026-04-17 10:54:30

80

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If $n\in\mathbb N;$ show that $\frac{^nC_0}{x}-\frac{^nC_1}{x+1}+\frac{^nC_2}{x+2}-...+(-1)^n\frac{^nC_n}{x+n}=\frac{n!}{x(x+1)(x+2)...(x+n)}$

Published on 17 Apr 2026 - 10:54

score 42 · Answer 4 · 2026-04-15 15:24:06

42

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Limit of the ratios of consecutive central binomial coefficient integers

Published on 15 Apr 2026 - 15:24

score 86 · Answer 5 · 2026-04-16 20:42:30

86

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On combinatoric equality $\sum_{i=1}^{n-1}{n\choose i} i^{i-1} (n-i)^{n-i-1}=2(n-1)n^{n-2}$

Published on 16 Apr 2026 - 20:42

score 69 · Answer 6 · 2026-04-14 20:16:34

69

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Summations of Products of Binomial Coefficients

Published on 14 Apr 2026 - 20:16

score 49 · Answer 7 · 2026-04-16 06:54:34

49

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Expression for product of binomial coefficients

Published on 16 Apr 2026 - 6:54

score 77 · Answer 8 · 2026-04-10 14:35:38

77

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Algebraic proof of binomial identity

Published on 10 Apr 2026 - 14:35

score 58 · Answer 9 · 2026-04-12 10:58:19

58

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cobinatorical proof for a binomial identity

Published on 12 Apr 2026 - 10:58

score 56 · Answer 10 · 2026-04-16 14:21:26

Proof of the combinatorial identity: $\frac{k+1}{k+1-i} {k+1-i \choose i} = \frac{k}{k-i} {k-i \choose i} + \frac{k-1}{k-i} {k-i-1 \choose i-1}$

score 109 · Answer 11 · 2026-04-15 22:36:05

109

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To prove: $\sum_{r=0}^{n}{{(-1)^r}{n\choose r}(n-r)^k}=\begin{cases}0&;& k\lt n\\n!&;& k=n\end{cases}$

Published on 15 Apr 2026 - 22:36

score 67 · Answer 12 · 2026-04-12 20:40:41

67

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Solve this summation for a closed form

Published on 12 Apr 2026 - 20:40

score 216 · Answer 13 · 2026-04-14 04:46:11

216

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Is there a formula for $\sum_{j=1}^n \binom{m j}{m}$, where $m$ is an integer?

Published on 14 Apr 2026 - 4:46

score 224 · Answer 14 · 2026-04-14 16:42:51

224

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How to calculate $\sum_{k=1}^n \frac{\binom{n}{k} x^k(1-x)^{n-k}}{k}$

Published on 14 Apr 2026 - 16:42

score 525 · Answer 15 · 2026-04-16 07:40:59

525

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Using de Moivre's Theorem, show that: $\cos(5\theta)=\cos^5\theta-10 \cos^3\theta \sin^2\theta+5 \cos\theta \sin^4\theta$

Published on 16 Apr 2026 - 7:40

List Question

How to prove that $\sum_{j=0}^{k}{(-1)^j\binom{n}{j}}=(-1)^k\binom{n-1}{k}$?

Proving a summation series involving binomial coefficients

If $n\in\mathbb N;$ show that $\frac{^nC_0}{x}-\frac{^nC_1}{x+1}+\frac{^nC_2}{x+2}-...+(-1)^n\frac{^nC_n}{x+n}=\frac{n!}{x(x+1)(x+2)...(x+n)}$

Limit of the ratios of consecutive central binomial coefficient integers

On combinatoric equality $\sum_{i=1}^{n-1}{n\choose i} i^{i-1} (n-i)^{n-i-1}=2(n-1)n^{n-2}$

Summations of Products of Binomial Coefficients

Expression for product of binomial coefficients

Algebraic proof of binomial identity

cobinatorical proof for a binomial identity

Proof of the combinatorial identity: $\frac{k+1}{k+1-i} {k+1-i \choose i} = \frac{k}{k-i} {k-i \choose i} + \frac{k-1}{k-i} {k-i-1 \choose i-1}$

To prove: $\sum_{r=0}^{n}{{(-1)^r}{n\choose r}(n-r)^k}=\begin{cases}0&;& k\lt n\\n!&;& k=n\end{cases}$

Solve this summation for a closed form

Is there a formula for $\sum_{j=1}^n \binom{m j}{m}$, where $m$ is an integer?

How to calculate $\sum_{k=1}^n \frac{\binom{n}{k} x^k(1-x)^{n-k}}{k}$

Using de Moivre's Theorem, show that: $\cos(5\theta)=\cos^5\theta-10 \cos^3\theta \sin^2\theta+5 \cos\theta \sin^4\theta$

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