Question

Combinatorial proof of identity $\sum_{k=0}^{\min(a,b)}\binom{x+y+k}{k}\binom{x}{b-k}\binom{y}{a-k} = \binom{x+a}{b}\binom{y+b}{a}$

score 432 · Answer 1 · 2015-09-08 09:22:45

432

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Combinatorial proof of identity $\sum_{k=0}^{\min(a,b)}\binom{x+y+k}{k}\binom{x}{b-k}\binom{y}{a-k} = \binom{x+a}{b}\binom{y+b}{a}$

Published on 08 Sep 2015 - 9:22

score 5.3k · Answer 2 · 2015-09-14 04:37:07

5.3k

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prove the combinatorial identity: $ {2n \choose n} + {2n\choose n-1} = (1/2){2n + 2 \choose n + 1}$

Published on 14 Sep 2015 - 4:37

score 11.8k · Answer 3 · 2015-09-16 15:55:12

11.8k

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Permutation of multiset

Published on 16 Sep 2015 - 15:55

score 99 · Answer 4 · 2015-09-16 16:09:57

99

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Show that $\binom{n}{k}$ equals to $\binom{n-1}{k}\cdot\frac{n-k+1}{k}$.

Published on 16 Sep 2015 - 16:09

score 138 · Answer 5 · 2015-09-21 14:13:12

138

Views

Discrete math combinatorial proof

Published on 21 Sep 2015 - 14:13

score 186 · Answer 6 · 2015-09-21 16:00:43

186

Views

Combinatorial Proof of $Q_n = D_n + D_{n-1}$

Published on 21 Sep 2015 - 16:00

score 190 · Answer 7 · 2015-09-23 03:32:35

190

Views

Using Combinatorial proof to conclude $\binom{2n}{n} = 2 \binom{2n-1}{n-1}$

Published on 23 Sep 2015 - 3:32

score 204 · Answer 8 · 2026-03-26 12:53:58

204

Views

Give a combinatorial proof of $\left(\!\!\binom{n}{k - n}\!\!\right) = {k - 1 \choose k - n}$

Published on 26 Mar 2026 - 12:53

score 275 · Answer 9 · 2015-09-27 02:36:50

275

Views

Prove that $\binom nk=\sum_{j=0}^{\lfloor\frac k2\rfloor}(-1)^j\binom nj\binom{n+k-2j-1}{n-1}$

Published on 27 Sep 2015 - 2:36

score 99 · Answer 10 · 2015-10-01 01:52:44

99

Views

Prove $\sum_{k=0}^n\binom{2n+1}{2k}=4^n$

Published on 01 Oct 2015 - 1:52

score 339 · Answer 11 · 2015-10-07 10:31:24

339

Views

Seeking a combinatorial proof $\sum\limits _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$

Published on 07 Oct 2015 - 10:31

score 777 · Answer 12 · 2015-10-20 08:23:52

777

Views

Prove using a combinatorial argument the following statement: $\binom{n+m}{2} = \binom{m}{2} + \binom{n}{2} + \binom{n}{1}\binom{m}{1}$

Published on 20 Oct 2015 - 8:23

score 80 · Answer 13 · 2026-03-14 05:10:33

80

Views

Can we simplify this factorial summation $\sum_{i=0}^{n-1} i!(n-i)!$?

Published on 14 Mar 2026 - 5:10

score 348 · Answer 14 · 2026-03-22 10:07:47

348

Views

Deduce formula for $\sum_{j=0}^m {m \choose j}(-1)^j j^{m+1}$

Published on 22 Mar 2026 - 10:07

score 3.2k · Answer 15 · 2015-11-06 08:21:14

3.2k

Views

Prove that if four numbers are chosen from the set $\{1,2,3,4,5,6\}$, at least one pair must add up to $7$.

Published on 06 Nov 2015 - 8:21

List Question

Combinatorial proof of identity $\sum_{k=0}^{\min(a,b)}\binom{x+y+k}{k}\binom{x}{b-k}\binom{y}{a-k} = \binom{x+a}{b}\binom{y+b}{a}$

prove the combinatorial identity: $ {2n \choose n} + {2n\choose n-1} = (1/2){2n + 2 \choose n + 1}$

Permutation of multiset

Show that $\binom{n}{k}$ equals to $\binom{n-1}{k}\cdot\frac{n-k+1}{k}$.

Discrete math combinatorial proof

Combinatorial Proof of $Q_n = D_n + D_{n-1}$

Using Combinatorial proof to conclude $\binom{2n}{n} = 2 \binom{2n-1}{n-1}$

Give a combinatorial proof of $\left(\!\!\binom{n}{k - n}\!\!\right) = {k - 1 \choose k - n}$

Prove that $\binom nk=\sum_{j=0}^{\lfloor\frac k2\rfloor}(-1)^j\binom nj\binom{n+k-2j-1}{n-1}$

Prove $\sum_{k=0}^n\binom{2n+1}{2k}=4^n$

Seeking a combinatorial proof $\sum\limits _{k=0}^n (n-2k)^2\binom{n}{k}=n\times 2^n$

Prove using a combinatorial argument the following statement: $\binom{n+m}{2} = \binom{m}{2} + \binom{n}{2} + \binom{n}{1}\binom{m}{1}$

Can we simplify this factorial summation $\sum_{i=0}^{n-1} i!(n-i)!$?

Deduce formula for $\sum_{j=0}^m {m \choose j}(-1)^j j^{m+1}$

Prove that if four numbers are chosen from the set $\{1,2,3,4,5,6\}$, at least one pair must add up to $7$.

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