Question

Combinatorial proof that $(n-r){n+r-1 \choose r}{n \choose r} = n{n+r-1 \choose 2r}{2r \choose r}$

score 200 · Answer 1 · 2016-02-07 19:33:05

200

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Combinatorial proof that $(n-r){n+r-1 \choose r}{n \choose r} = n{n+r-1 \choose 2r}{2r \choose r}$

Published on 07 Feb 2016 - 19:33

score 3.6k · Answer 2 · 2016-02-16 21:45:53

3.6k

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n(n+1)/2 combinatorial proof (details in description)

Published on 16 Feb 2016 - 21:45

score 617 · Answer 3 · 2016-02-17 01:42:34

617

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Prove that $\sum\limits_{k=0}^m\binom{m}k\binom{n}{r+k}=\binom{m+n}{m+r}$ using combinatoric arguments.

Published on 17 Feb 2016 - 1:42

score 555 · Answer 4 · 2016-02-21 08:16:48

555

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Proof through combinatorial argument for $\sum_{i=1}^n (i-1)(n-i) = \binom{n}{3}$

Published on 21 Feb 2016 - 8:16

score 270 · Answer 5 · 2016-02-21 17:05:10

270

Views

Proving through combinatorial argument $\sum_{i=0}^k \binom{m+k-i-1}{k-i}\binom{n+i-1}{i} = \binom{m+n+k-1}{k}$

Published on 21 Feb 2016 - 17:05

score 508 · Answer 6 · 2016-03-01 15:21:39

508

Views

Does the functional equation $p(x^2)=p(x)p(x+1)$ have a combinatorial interpretation?

Published on 01 Mar 2016 - 15:21

score 659 · Answer 7 · 2016-03-01 23:05:57

659

Views

Use the "weirdo" method to explain why the amounts of subsets with even/odd elements are equal.

Published on 01 Mar 2016 - 23:05

score 396 · Answer 8 · 2026-03-26 19:00:37

396

Views

How to prove with double counting technique that $1+2+\dots+2^n=2^{n+1} -1$?

Published on 26 Mar 2026 - 19:00

score 1.2k · Answer 9 · 2016-03-23 16:59:20

1.2k

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Two interview questions

Published on 23 Mar 2016 - 16:59

score 351 · Answer 10 · 2016-03-28 16:30:28

351

Views

Can anyone give a combinatorial proof of the identity ${n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+...+(n-m+1){m \choose m}={n+2 \choose m+2}$

Published on 28 Mar 2016 - 16:30

score 1.2k · Answer 11 · 2026-02-23 15:18:57

1.2k

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Combinatorial Proof of Derangement Identity $n!=\sum_{i=0}^n \binom{n}{n-i}D_i$

Published on 23 Feb 2026 - 15:18

score 132 · Answer 12 · 2016-04-06 11:20:51

132

Views

Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$

Published on 06 Apr 2016 - 11:20

score 149 · Answer 13 · 2016-04-07 07:45:08

149

Views

Combinatorial proof of an identity between restricted counts of permutations and derangements

Published on 07 Apr 2016 - 7:45

score 503 · Answer 14 · 2016-04-16 05:05:13

503

Views

A combinatorial proof of an identity involving Euler’s $ \phi $-function.

Published on 16 Apr 2016 - 5:05

score 809 · Answer 15 · 2016-04-17 16:26:38

Combinatorial proof of multiset coefficient identity: $\left(\!\left({n\atop k}\right)\!\right)= \left(\!\!\left({k+1\atop n-1}\right)\!\!\right)$

List Question

Combinatorial proof that $(n-r){n+r-1 \choose r}{n \choose r} = n{n+r-1 \choose 2r}{2r \choose r}$

n(n+1)/2 combinatorial proof (details in description)

Prove that $\sum\limits_{k=0}^m\binom{m}k\binom{n}{r+k}=\binom{m+n}{m+r}$ using combinatoric arguments.

Proof through combinatorial argument for $\sum_{i=1}^n (i-1)(n-i) = \binom{n}{3}$

Proving through combinatorial argument $\sum_{i=0}^k \binom{m+k-i-1}{k-i}\binom{n+i-1}{i} = \binom{m+n+k-1}{k}$

Does the functional equation $p(x^2)=p(x)p(x+1)$ have a combinatorial interpretation?

Use the "weirdo" method to explain why the amounts of subsets with even/odd elements are equal.

How to prove with double counting technique that $1+2+\dots+2^n=2^{n+1} -1$?

Two interview questions

Can anyone give a combinatorial proof of the identity ${n \choose m} + 2{n-1 \choose m}+3{n-2 \choose m}+...+(n-m+1){m \choose m}={n+2 \choose m+2}$

Combinatorial Proof of Derangement Identity $n!=\sum_{i=0}^n \binom{n}{n-i}D_i$

Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$

Combinatorial proof of an identity between restricted counts of permutations and derangements

A combinatorial proof of an identity involving Euler’s $ \phi $-function.

Combinatorial proof of multiset coefficient identity: $\left(\!\left({n\atop k}\right)\!\right)= \left(\!\!\left({k+1\atop n-1}\right)\!\!\right)$

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