List questions from binomial-theorem

Published on 23 Feb 2026 - 16:58

98

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Show $\int_0^1 x^n (1-x)^{N-n} dx = \left(\binom{N}{n} (N+1)\right)^{-1}$

Published on 22 May 2022 - 2:55

93

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Representing the generating function using binomial theorem.

Published on 24 May 2022 - 19:19

153

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Closed-form expression for $\sum_{j=0}^k \binom{n+j-1}{j} a^j b^{k-j}$?

Published on 30 May 2022 - 11:49

43

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Understanding summation with variables

Published on 03 Jun 2022 - 20:08

32

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Understanding binomial theorem with negative integer of n

Published on 24 Feb 2026 - 4:33

66

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Is there a way to generate a specific permutation of a list using an algortihm/equation that might accept a few inputs?

Published on 07 Jun 2022 - 22:50

45

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Problem with binomial series: $ \big( 1+ \frac{a^2}{x^2} \big)^{-\frac{1}{2}} $

Published on 08 Jun 2022 - 21:50

93

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Prove by induction that $(n - 1)^k \le n^k$

Published on 10 Jun 2022 - 10:57

48

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Sum of binomial coefficients for a specific sum

Published on 12 Jun 2022 - 11:45

97

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Can we use $(1+x)^n = 1+nx$ where $x\to0$ and $n$ is $1/0$?

Published on 13 Jun 2022 - 15:42

188

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Find $X/1430$ when $X=(^{10}C_1)^2+2(^{10}C_2)^2+3(^{10}C_3)^2+ ...+10(^{10}C_{10})^2$

Published on 13 Jun 2022 - 15:57

26

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Proof by shifting down the upper limit in Pascal's identity

Published on 18 Jun 2022 - 0:09

107

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Nested Binomial coefficients

Published on 18 Jun 2022 - 9:14

183

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Binomial expansion where $n$ is unknown

Published on 23 Jun 2022 - 10:59

List Question

When $\text{ord}_n(p)/m = \text{ord}_{n/m}(p)=1$

Show $\int_0^1 x^n (1-x)^{N-n} dx = \left(\binom{N}{n} (N+1)\right)^{-1}$

Representing the generating function using binomial theorem.

Closed-form expression for $\sum_{j=0}^k \binom{n+j-1}{j} a^j b^{k-j}$?

Understanding summation with variables

Understanding binomial theorem with negative integer of n

Is there a way to generate a specific permutation of a list using an algortihm/equation that might accept a few inputs?

Problem with binomial series: $ \big( 1+ \frac{a^2}{x^2} \big)^{-\frac{1}{2}} $

Prove by induction that $(n - 1)^k \le n^k$

Sum of binomial coefficients for a specific sum

Can we use $(1+x)^n = 1+nx$ where $x\to0$ and $n$ is $1/0$?

Find $X/1430$ when $X=(^{10}C_1)^2+2(^{10}C_2)^2+3(^{10}C_3)^2+ ...+10(^{10}C_{10})^2$

Proof by shifting down the upper limit in Pascal's identity

Nested Binomial coefficients

Binomial expansion where $n$ is unknown

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