Question

How can I prove that $\left|\sum_{i=0}^r (-1)^i \binom{a}{i} \binom{n-a}{r-i}\right| = \binom{n}{r}$, only for $a=0$ and $a=n$?

score 76 · Answer 1 · 2026-03-26 02:57:48

76

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How can I prove that $\left|\sum_{i=0}^r (-1)^i \binom{a}{i} \binom{n-a}{r-i}\right| = \binom{n}{r}$, only for $a=0$ and $a=n$?

Published on 26 Mar 2026 - 2:57

score 109 · Answer 2 · 2016-08-19 17:51:29

109

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Prove that if $n+1$ divides $\binom{2^{n}}{2}$ then $n+1$ is prime

Published on 19 Aug 2016 - 17:51

score 74 · Answer 3 · 2016-08-19 20:41:19

74

Views

If $7$ Divides $\binom{2^n}{2}+1$ then $n =3k+2$ for some positive integer $k$

Published on 19 Aug 2016 - 20:41

score 1.5k · Answer 4 · 2016-08-20 05:58:03

1.5k

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Show that $(1+x)^n$ = .... Hence deduce that .... = $2^n$

Published on 20 Aug 2016 - 5:58

score 1.2k · Answer 5 · 2026-03-31 05:58:18

1.2k

Views

sum of even binomial coefficients

Published on 31 Mar 2026 - 5:58

score 52 · Answer 6 · 2026-03-31 06:05:55

52

Views

Proving an equality with binomials

Published on 31 Mar 2026 - 6:05

score 122 · Answer 7 · 2026-03-31 06:00:28

122

Views

Coefficient of $x^{-2}$ in expression $(x-1/x)^{12}$

Published on 31 Mar 2026 - 6:00

score 166 · Answer 8 · 2016-08-20 19:19:55

166

Views

Solve $\binom{x+2}{5} = 126$ for $x$

Published on 20 Aug 2016 - 19:19

score 111 · Answer 9 · 2026-03-26 07:59:24

111

Views

Two conjectures about the diameter of the generalized Hamming graphs on distance $r\geq 1$.

Published on 26 Mar 2026 - 7:59

score 48 · Answer 10 · 2016-08-22 15:43:38

48

Views

Show that if $q$ is an odd prime then $3$ divides $\binom{2^q}{2}-1$ part 1

Published on 22 Aug 2016 - 15:43

score 59 · Answer 11 · 2016-08-22 16:10:09

59

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Show that if $q$ is an odd prime there does not exist a positive integer $n>5$ such that $3^n$ divides $\binom{2^q}{2}-1$

Published on 22 Aug 2016 - 16:10

score 89 · Answer 12 · 2026-04-01 01:20:22

89

Views

Closed Form For Summation

Published on 01 Apr 2026 - 1:20

score 186 · Answer 13 · 2016-08-23 14:27:55

186

Views

Prove $z^n + 1/{z^n}$ is rational if $z+1/z$ is rational

Published on 23 Aug 2016 - 14:27

score 2.9k · Answer 14 · 2026-03-25 01:20:24

2.9k

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Prove that $\sum\limits_{j=k}^n\,(-1)^{j-k}\,\binom{j}{k}\,\binom{2n-j}{j}\,2^{2(n-j)}=\binom{2n+1}{2k+1}$.

Published on 25 Mar 2026 - 1:20

score 260 · Answer 15 · 2026-03-25 01:18:47

260

Views

Inverse of the sum $\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$

Published on 25 Mar 2026 - 1:18

List Question

How can I prove that $\left|\sum_{i=0}^r (-1)^i \binom{a}{i} \binom{n-a}{r-i}\right| = \binom{n}{r}$, only for $a=0$ and $a=n$?

Prove that if $n+1$ divides $\binom{2^{n}}{2}$ then $n+1$ is prime

If $7$ Divides $\binom{2^n}{2}+1$ then $n =3k+2$ for some positive integer $k$

Show that $(1+x)^n$ = .... Hence deduce that .... = $2^n$

sum of even binomial coefficients

Proving an equality with binomials

Coefficient of $x^{-2}$ in expression $(x-1/x)^{12}$

Solve $\binom{x+2}{5} = 126$ for $x$

Two conjectures about the diameter of the generalized Hamming graphs on distance $r\geq 1$.

Show that if $q$ is an odd prime then $3$ divides $\binom{2^q}{2}-1$ part 1

Show that if $q$ is an odd prime there does not exist a positive integer $n>5$ such that $3^n$ divides $\binom{2^q}{2}-1$

Closed Form For Summation

Prove $z^n + 1/{z^n}$ is rational if $z+1/z$ is rational

Prove that $\sum\limits_{j=k}^n\,(-1)^{j-k}\,\binom{j}{k}\,\binom{2n-j}{j}\,2^{2(n-j)}=\binom{2n+1}{2k+1}$.

Inverse of the sum $\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$

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