Question

What is the combinatorial interpretation for this expression$ \sum_{i=0}^{n} (-1)^{i} \binom{n}{i} = 0$

score 70 · Answer 1 · 2021-03-03 10:04:35

70

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What is the combinatorial interpretation for this expression$ \sum_{i=0}^{n} (-1)^{i} \binom{n}{i} = 0$

Published on 03 Mar 2021 - 10:04

score 81 · Answer 2 · 2021-03-07 01:49:48

81

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Using binomial theorem to express $ \frac{1}{\sqrt{5}} \ [((\sqrt{5}/2)+(1/2))^{10} - (-(\sqrt{5}/2)+(1/2))^{10}]$ as a single finite series

Published on 07 Mar 2021 - 1:49

score 33 · Answer 3 · 2021-03-07 03:33:30

33

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How to get more information about generalized binomial series

Published on 07 Mar 2021 - 3:33

score 210 · Answer 4 · 2021-03-07 11:02:46

210

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Prove that a certain sum of binomial coefficients is divisible by a power of 2

Published on 07 Mar 2021 - 11:02

score 242 · Answer 5 · 2021-03-07 16:41:33

242

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Binomial expansion on quadratic solution

Published on 07 Mar 2021 - 16:41

score 137 · Answer 6 · 2021-03-08 19:49:01

137

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An honest die is rolled 5 times. What is the probability that you will roll NO 6's?

Published on 08 Mar 2021 - 19:49

score 75 · Answer 7 · 2021-03-08 21:36:18

75

Views

How to prove that $a_n =(1+\frac1n)^n$ for all $n\ge 1$ applies to $a_n=\sum_{k=0}^n \frac1{k!}(1-\frac1n)(1-\frac2n)\cdots(1-\frac{k-1}n)$?

Published on 08 Mar 2021 - 21:36

score 48 · Answer 8 · 2021-03-13 13:52:37

48

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Explanation for $k{n \choose k - 1} = (k-1){n \choose k - 1} + {n \choose k - 1}$

Published on 13 Mar 2021 - 13:52

score 66 · Answer 9 · 2021-03-14 06:00:39

66

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If $x^n=a_0+a_1(1+x)+\ldots+a_n(1+x)^n=b_0 + b_1(1-x)+\ldots+b_n(1-x)^n$, then for $n=101$, find $(a_{50},b_{50})$

Published on 14 Mar 2021 - 6:00

score 775 · Answer 10 · 2026-04-03 12:58:06

775

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Binomial theorem from first principles?

Published on 03 Apr 2026 - 12:58

score 34 · Answer 11 · 2021-03-14 18:17:53

prove using binomial theorem $\left(1+\frac{1}{n}\right)^n = 1 + \sum^{n}_{k=1}\left[\frac{1}{k!}\prod^{k-1}_{r=0}\left(1-\frac{r}{n}\right)\right]$

score 108 · Answer 12 · 2021-03-16 04:24:03

108

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The coefficient of $x^2$ in the expansion of $\left(x^3+2x^2+x+4\right)^{15}$

Published on 16 Mar 2021 - 4:24

score 860 · Answer 13 · 2021-03-16 22:25:31

860

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Finding the coefficient of $x^2$ in an expansion

Published on 16 Mar 2021 - 22:25

score 117 · Answer 14 · 2021-03-17 22:10:01

117

Views

Newton's Binomial Theorem with $n<0$

Published on 17 Mar 2021 - 22:10

score 907 · Answer 15 · 2026-03-30 13:59:11

907

Views

Prove that $\sum_{m=1}^{n} (-1)^{m+1} {n \choose m} \frac{1}{m+1} = \frac{n}{n+1}$

Published on 30 Mar 2026 - 13:59

List Question

What is the combinatorial interpretation for this expression$ \sum_{i=0}^{n} (-1)^{i} \binom{n}{i} = 0$

Using binomial theorem to express $ \frac{1}{\sqrt{5}} \ [((\sqrt{5}/2)+(1/2))^{10} - (-(\sqrt{5}/2)+(1/2))^{10}]$ as a single finite series

How to get more information about generalized binomial series

Prove that a certain sum of binomial coefficients is divisible by a power of 2

Binomial expansion on quadratic solution

An honest die is rolled 5 times. What is the probability that you will roll NO 6's?

How to prove that $a_n =(1+\frac1n)^n$ for all $n\ge 1$ applies to $a_n=\sum_{k=0}^n \frac1{k!}(1-\frac1n)(1-\frac2n)\cdots(1-\frac{k-1}n)$?

Explanation for $k{n \choose k - 1} = (k-1){n \choose k - 1} + {n \choose k - 1}$

If $x^n=a_0+a_1(1+x)+\ldots+a_n(1+x)^n=b_0 + b_1(1-x)+\ldots+b_n(1-x)^n$, then for $n=101$, find $(a_{50},b_{50})$

Binomial theorem from first principles?

prove using binomial theorem $\left(1+\frac{1}{n}\right)^n = 1 + \sum^{n}_{k=1}\left[\frac{1}{k!}\prod^{k-1}_{r=0}\left(1-\frac{r}{n}\right)\right]$

The coefficient of $x^2$ in the expansion of $\left(x^3+2x^2+x+4\right)^{15}$

Finding the coefficient of $x^2$ in an expansion

Newton's Binomial Theorem with $n<0$

Prove that $\sum_{m=1}^{n} (-1)^{m+1} {n \choose m} \frac{1}{m+1} = \frac{n}{n+1}$

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