Question

The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is

score 272 · Answer 1 · 2014-04-03 13:25:26

272

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The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is

Published on 03 Apr 2014 - 13:25

score 431 · Answer 2 · 2014-04-03 13:33:37

431

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Upper bound for ${n \choose cn}$

Published on 03 Apr 2014 - 13:33

score 179 · Answer 3 · 2026-03-27 03:48:10

179

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Infinite series containing binomial coefficients

Published on 27 Mar 2026 - 3:48

score 3.1k · Answer 4 · 2014-04-05 03:32:36

3.1k

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coefficient of $x^2$, in $(1+x+x^2)^{10}$

Published on 05 Apr 2014 - 3:32

score 133 · Answer 5 · 2014-04-05 20:32:31

133

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Binomial coefficients inequality

Published on 05 Apr 2014 - 20:32

score 167 · Answer 6 · 2026-03-28 08:09:53

167

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How to find the coefficient of $x^8$ in $\prod\limits_{i=1}^{10}{\left(x-i\right)}$?

Published on 28 Mar 2026 - 8:09

score 1.4k · Answer 7 · 2014-04-06 16:34:21

1.4k

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Probability of two independent random variables being equal

Published on 06 Apr 2014 - 16:34

score 221 · Answer 8 · 2014-04-06 18:46:50

221

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Simplifying Sum of Subsets

Published on 06 Apr 2014 - 18:46

score 262 · Answer 9 · 2014-04-06 18:49:13

262

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Limit of $\sum_{k=0}^{\lfloor n/2 \rfloor} 2^{-2nk} \binom{n}{2k}\left(\binom{2k}{k}^n\right)$

Published on 06 Apr 2014 - 18:49

score 377 · Answer 10 · 2014-04-07 05:30:47

377

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Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$

Published on 07 Apr 2014 - 5:30

score 3.3k · Answer 11 · 2014-04-07 08:10:42

3.3k

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Prove $k\binom nk=n \binom{k-1}{ n-1}$ algebraically.

Published on 07 Apr 2014 - 8:10

score 121 · Answer 12 · 2026-03-27 23:00:54

121

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Closed form for $\prod_{k=1}^n \binom{k^2+2k}{k^2}$

Published on 27 Mar 2026 - 23:00

score 5k · Answer 13 · 2014-04-09 10:18:59

5k

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Taking Limits with Binomial Coefficients

Published on 09 Apr 2014 - 10:18

score 90 · Answer 14 · 2014-04-09 15:16:58

90

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What is $\lim_{n\to\infty} \sum_{k=1}^{n/2} \left(\frac{en}{2k}\right)^{2k} \frac{1}{\sqrt{\pi k}}^{\frac{3n}{\log_2{n}}}\,?$

Published on 09 Apr 2014 - 15:16

score 278 · Answer 15 · 2014-04-09 16:04:19

What is $\lim_{n\to\infty} \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$

List Question

The value of $\binom{50}0\binom{50}1+\binom{50}1\binom{50}2+\dots+\binom{50}{49}\binom{50}{50}$ is

Upper bound for ${n \choose cn}$

Infinite series containing binomial coefficients

coefficient of $x^2$, in $(1+x+x^2)^{10}$

Binomial coefficients inequality

How to find the coefficient of $x^8$ in $\prod\limits_{i=1}^{10}{\left(x-i\right)}$?

Probability of two independent random variables being equal

Simplifying Sum of Subsets

Limit of $\sum_{k=0}^{\lfloor n/2 \rfloor} 2^{-2nk} \binom{n}{2k}\left(\binom{2k}{k}^n\right)$

Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$

Prove $k\binom nk=n \binom{k-1}{ n-1}$ algebraically.

Closed form for $\prod_{k=1}^n \binom{k^2+2k}{k^2}$

Taking Limits with Binomial Coefficients

What is $\lim_{n\to\infty} \sum_{k=1}^{n/2} \left(\frac{en}{2k}\right)^{2k} \frac{1}{\sqrt{\pi k}}^{\frac{3n}{\log_2{n}}}\,?$

What is $\lim_{n\to\infty} \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$

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