Question

Counting argument for $\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}\binom{n}{j}\frac{1}{k} = \frac{1}{n}$

score 200 · Answer 1 · 2026-04-14 10:17:00

200

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Counting argument for $\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}\binom{n}{j}\frac{1}{k} = \frac{1}{n}$

Published on 14 Apr 2026 - 10:17

score 1.8k · Answer 2 · 2026-04-08 23:54:22

1.8k

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expanding a factorial by simplifying with $(k-1)!$

Published on 08 Apr 2026 - 23:54

score 203 · Answer 3 · 2026-04-10 10:00:55

203

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Derive a Combinatorial Identity

Published on 10 Apr 2026 - 10:00

score 294 · Answer 4 · 2026-04-13 07:36:50

294

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counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$

Published on 13 Apr 2026 - 7:36

score 173 · Answer 5 · 2026-04-17 14:27:16

173

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Number of zeroes and sign switches in a constructed zero-sum double sequence

Published on 17 Apr 2026 - 14:27

score 368 · Answer 6 · 2026-04-17 06:20:56

368

Views

proving the identity $b^3 = 6\binom{b}{3} + 6\binom{b}{2} + b$ for $b \in \mathbb{N}$, $b > 2$, by counting a set in different ways

Published on 17 Apr 2026 - 6:20

score 175 · Answer 7 · 2026-04-16 03:40:28

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 558 · Answer 8 · 2026-04-16 19:48:54

558

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Combinatorial proof for $2^n=1+\sum_{k=0}^{n-1}2^k$

Published on 16 Apr 2026 - 19:48

score 2.6k · Answer 9 · 2026-04-15 04:45:38

2.6k

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The properties of graph and its relation with the largest eigenvalue

Published on 15 Apr 2026 - 4:45

score 1.6k · Answer 10 · 2026-04-16 20:46:57

1.6k

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Prove $1^2+2^2+\cdots+n^2 = {n+1\choose2}+2{n+1\choose3}$

Published on 16 Apr 2026 - 20:46

score 83 · Answer 11 · 2026-04-12 04:07:04

83

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Confirming the correctness of Combinatoric permutation formula

Published on 12 Apr 2026 - 4:07

score 407 · Answer 12 · 2026-04-10 20:44:55

407

Views

Combinatorial proofs - how?

Published on 10 Apr 2026 - 20:44

score 300 · Answer 13 · 2026-04-10 16:01:56

300

Views

Finite sum identity involving Stirling numbers

Published on 10 Apr 2026 - 16:01

score 23.3k · Answer 14 · 2026-04-12 17:59:29

23.3k

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Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$

Published on 12 Apr 2026 - 17:59

score 405 · Answer 15 · 2026-04-13 00:39:23

405

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Combinatorial proof of $k\binom{n}{k} = n\binom{n-1}{k-1}$

Published on 13 Apr 2026 - 0:39

List Question

Counting argument for $\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}\binom{n}{j}\frac{1}{k} = \frac{1}{n}$

expanding a factorial by simplifying with $(k-1)!$

Derive a Combinatorial Identity

counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$

Number of zeroes and sign switches in a constructed zero-sum double sequence

proving the identity $b^3 = 6\binom{b}{3} + 6\binom{b}{2} + b$ for $b \in \mathbb{N}$, $b > 2$, by counting a set in different ways

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

Combinatorial proof for $2^n=1+\sum_{k=0}^{n-1}2^k$

The properties of graph and its relation with the largest eigenvalue

Prove $1^2+2^2+\cdots+n^2 = {n+1\choose2}+2{n+1\choose3}$

Confirming the correctness of Combinatoric permutation formula

Combinatorial proofs - how?

Finite sum identity involving Stirling numbers

Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$

Combinatorial proof of $k\binom{n}{k} = n\binom{n-1}{k-1}$

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