List questions from binomial-coefficients

Published on 30 Mar 2026 - 20:44

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Is there any closed form for $\sum_{k=0}^{n}k^{r}\binom{n}{k}$?

Published on 30 Mar 2026 - 20:43

105

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Provide a combinatorial proof for an identity using the Binomial Theorem

Published on 27 Mar 2026 - 0:01

121

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If the fractional part of the number $\dfrac{2^{403}}{15}$ is $\dfrac{k}{15}$, then $k$ is equal to

Published on 30 Mar 2026 - 0:58

268

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Find the power series expansion using the Generalized Binomial Theorem

Published on 30 Mar 2026 - 0:53

194

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sum of binomial series with alternate terms

Published on 22 Feb 2020 - 13:00

1.2k

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Poisson paradigm in "near-birthday problem" example

Published on 23 Feb 2026 - 11:37

23

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Formula to sum columns on Pascal's triangle

Published on 23 Feb 2020 - 14:41

70

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Prove that this sum of products of binomial coefficient can be simplified like this

Published on 23 Feb 2020 - 15:22

59

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Another way to express choosing $k$ objects from $n$

Published on 24 Feb 2020 - 0:27

257

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Calculate sum with binomials using generating functions

Published on 01 Apr 2026 - 14:19

83

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Evaluate $g(n,j)=\sum _{k=1}^n \frac{k^j (-1)^{n-k} \binom{n}{k}}{\frac{1}{2} n (n+1)-k}$

Published on 30 Mar 2026 - 20:44

135

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Evaluate $\sum _{j=0}^n \frac{\binom{n}{j}^2}{(j+1)^2}$

Published on 25 Feb 2020 - 6:23

32

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Clarification on final step of proof of binomial theorem by induction.

Published on 30 Mar 2026 - 0:58

1k

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$\exp(a)\exp(b) = \exp(a+b)$

Published on 31 Mar 2026 - 5:20

List Question

Evaluate $f(n,j)=\sum _{k=0}^n k^{2 j} (-1)^{n-k} \binom{2 n}{n-k}$

Is there any closed form for $\sum_{k=0}^{n}k^{r}\binom{n}{k}$?

Provide a combinatorial proof for an identity using the Binomial Theorem

If the fractional part of the number $\dfrac{2^{403}}{15}$ is $\dfrac{k}{15}$, then $k$ is equal to

Find the power series expansion using the Generalized Binomial Theorem

sum of binomial series with alternate terms

Poisson paradigm in "near-birthday problem" example

Formula to sum columns on Pascal's triangle

Prove that this sum of products of binomial coefficient can be simplified like this

Another way to express choosing $k$ objects from $n$

Calculate sum with binomials using generating functions

Evaluate $g(n,j)=\sum _{k=1}^n \frac{k^j (-1)^{n-k} \binom{n}{k}}{\frac{1}{2} n (n+1)-k}$

Evaluate $\sum _{j=0}^n \frac{\binom{n}{j}^2}{(j+1)^2}$

Clarification on final step of proof of binomial theorem by induction.

$\exp(a)\exp(b) = \exp(a+b)$

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