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What is the coefficient of the following

score 79 · Answer 1 · 2026-04-02 15:13:56

79

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What is the coefficient of the following

Published on 02 Apr 2026 - 15:13

score 1.2k · Answer 2 · 2016-07-09 17:52:38

1.2k

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Proof/derivation of $\lim\limits_{n\to\infty}{\frac1{2^n}\sum\limits_{k=0}^n\binom{n}{k}\frac{an+bk}{cn+dk}}\stackrel?=\frac{2a+b}{2c+d}$?

Published on 09 Jul 2016 - 17:52

score 111 · Answer 3 · 2016-07-11 17:56:47

111

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Number of routes

Published on 11 Jul 2016 - 17:56

score 244 · Answer 4 · 2016-07-12 12:28:17

244

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Coefficient of $x^{41}$ in $(x^5 + x^6 + x^7 + x^8 + x^9)^5$

Published on 12 Jul 2016 - 12:28

score 380 · Answer 5 · 2026-03-25 11:20:06

380

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Median poker hand (Texas Hold’em)

Published on 25 Mar 2026 - 11:20

score 59 · Answer 6 · 2026-03-31 00:01:52

If $f(n) =\displaystyle\sum_{r=1}^{n}\Biggl(r^n\Bigg(\binom{n}{r}-\binom{n}{r-1}\Bigg) + (2r+1)\binom{n}{r}\Biggr)$, then what is $f(30)$?

score 167 · Answer 7 · 2016-07-13 17:20:44

167

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Coefficient of $x^{r}$

Published on 13 Jul 2016 - 17:20

score 334 · Answer 8 · 2026-03-30 20:45:10

334

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About "The product of the six numbers surrounding any interior number in Pascal’s triangle is a perfect square"

Published on 30 Mar 2026 - 20:45

score 296 · Answer 9 · 2016-07-15 16:18:36

296

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How to prove: $\sum_{k=m+1}^{n} (-1)^{k} \binom{n}{k}\binom{k-1}{m}= (-1)^{m+1}$

Published on 15 Jul 2016 - 16:18

score 549 · Answer 10 · 2026-03-30 16:26:35

549

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Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?

Published on 30 Mar 2026 - 16:26

score 138 · Answer 11 · 2016-07-16 18:21:39

138

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Show that $\sum_{k=0}^n \frac{(2n)!}{{k!^2(n-k)!}^2}= \binom{2n}{n}^2$

Published on 16 Jul 2016 - 18:21

score 1.2k · Answer 12 · 2026-03-27 15:09:52

1.2k

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Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion

Published on 27 Mar 2026 - 15:09

score 211 · Answer 13 · 2016-07-18 07:24:00

211

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Prove that $\sum_{k=0}^n \binom{3n-k}{2n}=\binom{3n+1}{n}$

Published on 18 Jul 2016 - 7:24

score 144 · Answer 14 · 2016-07-18 14:36:26

144

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How to find number of integral solutions, containing large number of cases?

Published on 18 Jul 2016 - 14:36

score 1.2k · Answer 15 · 2016-07-18 14:43:10

1.2k

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Differentiating the binomial coefficient

Published on 18 Jul 2016 - 14:43

List Question

What is the coefficient of the following

Proof/derivation of $\lim\limits_{n\to\infty}{\frac1{2^n}\sum\limits_{k=0}^n\binom{n}{k}\frac{an+bk}{cn+dk}}\stackrel?=\frac{2a+b}{2c+d}$?

Number of routes

Coefficient of $x^{41}$ in $(x^5 + x^6 + x^7 + x^8 + x^9)^5$

Median poker hand (Texas Hold’em)

If $f(n) =\displaystyle\sum_{r=1}^{n}\Biggl(r^n\Bigg(\binom{n}{r}-\binom{n}{r-1}\Bigg) + (2r+1)\binom{n}{r}\Biggr)$, then what is $f(30)$?

Coefficient of $x^{r}$

About "The product of the six numbers surrounding any interior number in Pascal’s triangle is a perfect square"

How to prove: $\sum_{k=m+1}^{n} (-1)^{k} \binom{n}{k}\binom{k-1}{m}= (-1)^{m+1}$

Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?

Show that $\sum_{k=0}^n \frac{(2n)!}{{k!^2(n-k)!}^2}= \binom{2n}{n}^2$

Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion

Prove that $\sum_{k=0}^n \binom{3n-k}{2n}=\binom{3n+1}{n}$

How to find number of integral solutions, containing large number of cases?

Differentiating the binomial coefficient

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