Question

Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$

score 1.1k · Answer 1 · 2020-02-24 23:03:12

1.1k

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Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$

Published on 24 Feb 2020 - 23:03

score 89 · Answer 2 · 2020-02-25 01:07:36

89

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Combinatorial argument for Stirling number identity

Published on 25 Feb 2020 - 1:07

score 64 · Answer 3 · 2020-03-01 05:19:08

64

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Prove without Binomial theorem

Published on 01 Mar 2020 - 5:19

score 78 · Answer 4 · 2020-03-03 08:58:51

78

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Given $s,n,k$ such that $n\ge k+1$ prove the following: $\sum^s_{r=0} {n+r \choose k} = {n+s+1\choose k+1} - {n\choose k+1}$

Published on 03 Mar 2020 - 8:58

score 1k · Answer 5 · 2020-03-03 11:22:15

1k

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Combinatorial proof of identity involving central binomial coefficients $ \sum_{k=0}^{n-1} {2k \choose k} \cdot 2^{2(n-k)} = 2n \cdot {2n \choose n}$

Published on 03 Mar 2020 - 11:22

score 61 · Answer 6 · 2020-03-04 17:09:14

61

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Bijective Functions.

Published on 04 Mar 2020 - 17:09

score 180 · Answer 7 · 2026-03-26 02:11:58

180

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A game of tourism

Published on 26 Mar 2026 - 2:11

score 108 · Answer 8 · 2020-03-11 13:52:54

108

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Is this sum of rational numbers congruent to zero modulo p?

Published on 11 Mar 2020 - 13:52

score 53 · Answer 9 · 2020-03-11 15:15:41

53

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Closed form for $\sum_{p=0}^{i} \binom{a}{p} \binom{b}{i-p} \binom{b+c -(i-p)}{m-i}$

Published on 11 Mar 2020 - 15:15

score 260 · Answer 10 · 2020-03-14 15:13:39

260

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Is my proof by double counting correct?

Published on 14 Mar 2020 - 15:13

score 142 · Answer 11 · 2020-03-16 13:24:51

142

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Combinatoric interpretation of an equality between sums

Published on 16 Mar 2020 - 13:24

score 77 · Answer 12 · 2020-03-17 10:06:20

77

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Combinatoric proof of an equality

Published on 17 Mar 2020 - 10:06

score 715 · Answer 13 · 2020-03-19 10:59:12

715

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How to give a combinatorial proof for: If $n$ and $k$ are positive integers with $n=2k$ then $\frac{n!}{2^k}$ is an integer

Published on 19 Mar 2020 - 10:59

score 122 · Answer 14 · 2026-03-26 01:27:10

122

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The adjacent number game.

Published on 26 Mar 2026 - 1:27

score 33 · Answer 15 · 2020-03-21 16:05:24

Given r blocks of one color and g blocks of another color a tower is made. Show that the max height of the tower is independent of the colors.

List Question

Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$

Combinatorial argument for Stirling number identity

Prove without Binomial theorem

Given $s,n,k$ such that $n\ge k+1$ prove the following: $\sum^s_{r=0} {n+r \choose k} = {n+s+1\choose k+1} - {n\choose k+1}$

Combinatorial proof of identity involving central binomial coefficients $ \sum_{k=0}^{n-1} {2k \choose k} \cdot 2^{2(n-k)} = 2n \cdot {2n \choose n}$

Bijective Functions.

A game of tourism

Is this sum of rational numbers congruent to zero modulo p?

Closed form for $\sum_{p=0}^{i} \binom{a}{p} \binom{b}{i-p} \binom{b+c -(i-p)}{m-i}$

Is my proof by double counting correct?

Combinatoric interpretation of an equality between sums

Combinatoric proof of an equality

How to give a combinatorial proof for: If $n$ and $k$ are positive integers with $n=2k$ then $\frac{n!}{2^k}$ is an integer

The adjacent number game.

Given r blocks of one color and g blocks of another color a tower is made. Show that the max height of the tower is independent of the colors.

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