Question

Prove that $\sum_{k=0}^n (-1)^k\frac{{ {n}\choose{k}}{ {x}\choose{k}}}{{ {y}\choose{k}}} = \frac{{ {y-x}\choose{n}}}{{ {y}\choose{n}}}$

score 379 · Answer 1 · 2016-12-19 02:12:47

379

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Prove that $\sum_{k=0}^n (-1)^k\frac{{ {n}\choose{k}}{ {x}\choose{k}}}{{ {y}\choose{k}}} = \frac{{ {y-x}\choose{n}}}{{ {y}\choose{n}}}$

Published on 19 Dec 2016 - 2:12

score 444 · Answer 2 · 2016-12-19 10:40:57

444

Views

How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$

Published on 19 Dec 2016 - 10:40

score 124 · Answer 3 · 2026-03-25 21:54:06

124

Views

Show that $-{C_{x-2}\over {2x-1\choose x}}=1-{x-1\over x+1}(2^2)+{(x-1)(x-2)\over (x+1)(x+2) }(3^2)-\cdots$

Published on 25 Mar 2026 - 21:54

score 1k · Answer 4 · 2026-04-12 01:35:44

1k

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Understanding how to use the binomial theorem to expand $(x+y)^3$

Published on 12 Apr 2026 - 1:35

score 1.2k · Answer 5 · 2016-12-19 22:19:16

1.2k

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If the coefficients of $x^{k}$ and $x^{k+1}$ in the expansion of $(2+3x)^{19}$ are equal, how to find $k$?

Published on 19 Dec 2016 - 22:19

score 206 · Answer 6 · 2026-04-12 00:47:29

206

Views

Proving that a hard sum equals 1

Published on 12 Apr 2026 - 0:47

score 200 · Answer 7 · 2026-04-04 17:34:07

200

Views

Common proof for $(1+x)(1+x^2)(1+x^4)...(1+x^{2^n})=\dfrac{1-x^{2^{n+1}}}{1-x} $

Published on 04 Apr 2026 - 17:34

score 1.4k · Answer 8 · 2026-03-30 16:02:29

1.4k

Views

GCD of binomial coefficients

Published on 30 Mar 2026 - 16:02

score 5.6k · Answer 9 · 2026-03-26 03:12:47

5.6k

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Finding the coefficient of $x^{10}$ in $(1+x^2-x^3)^8$

Published on 26 Mar 2026 - 3:12

score 512 · Answer 10 · 2026-04-13 00:13:17

512

Views

Sum of product of squared binomial coefficients

Published on 13 Apr 2026 - 0:13

score 192 · Answer 11 · 2026-04-13 07:12:59

192

Views

Prove without induction that $\sum_{r=1}^n {n\choose r}(-1)^{r+1}\frac1r=\sum_{r=1}^n \frac1r$

Published on 13 Apr 2026 - 7:12

score 252 · Answer 12 · 2016-12-22 21:51:40

252

Views

How to prove $\int_{-\infty}^{\infty}\binom{n}{x}^2\binom{2n+x}{x} dx=\binom{2n}{n}^2$

Published on 22 Dec 2016 - 21:51

score 171 · Answer 13 · 2016-12-22 22:47:59

171

Views

Proof that $\sum_{k=0}^n \binom{k}{p}$ = $\binom{n+1}{p+1}$

Published on 22 Dec 2016 - 22:47

score 89 · Answer 14 · 2026-03-29 10:55:55

89

Views

Approximating $\binom{N}{N/2}$

Published on 29 Mar 2026 - 10:55

score 91 · Answer 15 · 2026-04-11 00:31:48

91

Views

Building a single likelihood for the sum of two events

Published on 11 Apr 2026 - 0:31

List Question

Prove that $\sum_{k=0}^n (-1)^k\frac{{ {n}\choose{k}}{ {x}\choose{k}}}{{ {y}\choose{k}}} = \frac{{ {y-x}\choose{n}}}{{ {y}\choose{n}}}$

How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$

Show that $-{C_{x-2}\over {2x-1\choose x}}=1-{x-1\over x+1}(2^2)+{(x-1)(x-2)\over (x+1)(x+2) }(3^2)-\cdots$

Understanding how to use the binomial theorem to expand $(x+y)^3$

If the coefficients of $x^{k}$ and $x^{k+1}$ in the expansion of $(2+3x)^{19}$ are equal, how to find $k$?

Proving that a hard sum equals 1

Common proof for $(1+x)(1+x^2)(1+x^4)...(1+x^{2^n})=\dfrac{1-x^{2^{n+1}}}{1-x} $

GCD of binomial coefficients

Finding the coefficient of $x^{10}$ in $(1+x^2-x^3)^8$

Sum of product of squared binomial coefficients

Prove without induction that $\sum_{r=1}^n {n\choose r}(-1)^{r+1}\frac1r=\sum_{r=1}^n \frac1r$

How to prove $\int_{-\infty}^{\infty}\binom{n}{x}^2\binom{2n+x}{x} dx=\binom{2n}{n}^2$

Proof that $\sum_{k=0}^n \binom{k}{p}$ = $\binom{n+1}{p+1}$

Approximating $\binom{N}{N/2}$

Building a single likelihood for the sum of two events

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