Question

Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$

score 13.5k · Answer 1 · 2011-12-28 16:32:37

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Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$

Published on 28 Dec 2011 - 16:32

score 11.1k · Answer 2 · 2011-12-29 22:09:53

11.1k

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Combinatorial interpretation of sum of squares, cubes

Published on 29 Dec 2011 - 22:09

score 684 · Answer 3 · 2012-04-03 09:57:04

684

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Combinatorial Proof for a $ p\mid\binom{p}{k} \ \ \ \ \ 0<k<p$ .

Published on 03 Apr 2012 - 9:57

score 3k · Answer 4 · 2012-04-27 17:16:07

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Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially

Published on 27 Apr 2012 - 17:16

score 591 · Answer 5 · 2012-05-07 19:05:55

591

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Proof of a Binomial Identity using a combinatorial argument: $\binom{2k}{2} =\binom{k−l}{2}+ \binom{k+l}{2}+ k^2 − l^2$

Published on 07 May 2012 - 19:05

score 104.1k · Answer 6 · 2012-05-23 04:52:43

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Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$

Published on 23 May 2012 - 4:52

score 2k · Answer 7 · 2012-09-19 19:12:07

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Combinatorially prove that $\sum_{i=0}^n {n \choose i} 2^i = 3^n $

Published on 19 Sep 2012 - 19:12

score 6.9k · Answer 8 · 2012-09-21 02:35:45

6.9k

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Combinatorial proof of the identity $\sum\limits_{i=0}^{n}{\binom{n}{i}}{2^i}=3^n$

Published on 21 Sep 2012 - 2:35

score 377 · Answer 9 · 2012-09-24 14:18:37

377

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Combinatorial (non-algebraic) Proof: $ {}_{n}C_{4}={}_{n-1}C_{3}$ +${}_{n-2}C_{3} +{}_{n-3}C_{3} + \cdots + {}_{3}C_{3} $

Published on 24 Sep 2012 - 14:18

score 15.6k · Answer 10 · 2026-02-23 18:45:37

15.6k

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Combinatorial argument to prove the recurrence relation for number of derangements

Published on 23 Feb 2026 - 18:45

score 6.5k · Answer 11 · 2012-11-05 02:58:05

6.5k

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How to begin combinatorial proof of $\sum_{k=1}^n k \binom nk^2 = n \binom{2n-1}{n-1}$

Published on 05 Nov 2012 - 2:58

score 2k · Answer 12 · 2013-02-11 03:59:57

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Help with combinatorial proof of binomial identity: $\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$

Published on 11 Feb 2013 - 3:59

score 618 · Answer 13 · 2013-02-18 03:50:47

618

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Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$

Published on 18 Feb 2013 - 3:50

score 2k · Answer 14 · 2013-02-25 11:18:51

2k

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A combinatorial proof that the alternating sum of binomial coefficients is zero

Published on 25 Feb 2013 - 11:18

score 1.5k · Answer 15 · 2026-03-26 22:14:07

1.5k

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Combinatorial proof for identity $\left(\!\!\binom{n\vphantom{1}}{k}\!\!\right)=\left(\!\!\binom{k+1}{n-1}\!\!\right)$ (multiset coefficients)

Published on 26 Mar 2026 - 22:14

List Question

Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$

Combinatorial interpretation of sum of squares, cubes

Combinatorial Proof for a $ p\mid\binom{p}{k} \ \ \ \ \ 0<k<p$ .

Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially

Proof of a Binomial Identity using a combinatorial argument: $\binom{2k}{2} =\binom{k−l}{2}+ \binom{k+l}{2}+ k^2 − l^2$

Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$

Combinatorially prove that $\sum_{i=0}^n {n \choose i} 2^i = 3^n $

Combinatorial proof of the identity $\sum\limits_{i=0}^{n}{\binom{n}{i}}{2^i}=3^n$

Combinatorial (non-algebraic) Proof: $ {}_{n}C_{4}={}_{n-1}C_{3}$ +${}_{n-2}C_{3} +{}_{n-3}C_{3} + \cdots + {}_{3}C_{3} $

Combinatorial argument to prove the recurrence relation for number of derangements

How to begin combinatorial proof of $\sum_{k=1}^n k \binom nk^2 = n \binom{2n-1}{n-1}$

Help with combinatorial proof of binomial identity: $\sum\limits_{k=1}^nk^2{n\choose k}^2 = n^2{2n-2\choose n-1}$

Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$

A combinatorial proof that the alternating sum of binomial coefficients is zero

Combinatorial proof for identity $\left(\!\!\binom{n\vphantom{1}}{k}\!\!\right)=\left(\!\!\binom{k+1}{n-1}\!\!\right)$ (multiset coefficients)

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