Questions tagged [binomial-theorem]

For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.

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Representing the generating function using binomial theorem.

Lets say we have a set {1, 2, 3, 4, 5}, There are 32 subsets. Now if we take these subsets and arrange them according to the sum of their members; For eg: The subset {2, 3} would be in the same group ...
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Prove $(a+b)^{p} \leq a^{p}+b^{p}$ if $a,b>0$ and $p \in (0,1)$ [duplicate]

I need to prove that if $a,b>0$ and $p \in (0,1)$ then $(a+b)^{p} \leq a^{p}+b^{p}$. I've been trying to use the generalized binomial theorem but i haven't solve it yet.
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quickest way of determining the the coefficient of any term of a binomial expansion

What are the coefficients of $x^4$ and $x^5$ of this binomial $(1+x)\left(1-\frac x2\right)^8$? and also share the quickest way of solving this type of math (without expanding it fully if possible)!
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Given $p$ red balls and $q$ blue balls, in how many ways can I pick 2 red, then 2 blue, then 2 red, in a row?

Arranging $p$ red balls and $q$ blue balls on a single line, in how many ways can I put 2 red, then 2 blue, then 2 red, in a row? Even further, in how many ways can I pick 2 reds 2 blues 2 reds in a ...
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Given the expression $$P_n(x)=\dfrac{1}{2^n}\left[\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n \right]$$ Prove that $P_n(x)$ satisfies the identity $$P_n(x)-xP_{n-1}(x)+\frac{1}{4}P_{n-... • 1,563 -1 votes 1 answer 70 views Show that \sum_{i=0}^{n}\binom{k}{i}=2^{k-1} [duplicate] Show that$$\sum_{i=0}^{n}\binom{k}{i}=2^{k-1}$$for every positive odd integer in the form of k=2n+1 (n being a positive integer). I tried the usual induction demonstration. The case for k=1 is ... 1 vote 3 answers 68 views Binomial expansion of (1+x)^i where i^2 = -1. I was reading today about this single variable binomial expansion (1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\ldots For example, (1+x)^2 = 1+2x+x^2 However, is it also valid when ... • 805 1 vote 1 answer 71 views Proving divisibility by induction [duplicate] The problem consists of demonstrating through induction that \forall a \in \mathbb{N} and n\geq 0 (where n is an integer)$$(a^2+a+1)\mid(a^{n+2} +(a+1)^{2n+1})$$That is, a^2+a+1 divides a^{... • 39 0 votes 1 answer 55 views Prove that (\frac{13}{16})^{12}=\sum_{k=0}^{12}{12 \choose k}\left(\frac{1}{4}\right)^{2k}\left(\frac{3}{4}\right)^{12-k} [closed] I tried to solve a probability question where I got \sum_{k=0}^{12}{12 \choose k}\left(\frac{1}{4}\right)^{2k}\left(\frac{3}{4}\right)^{12-k} but did not succeed in evaluating that to (\frac{13}{16}... 3 votes 3 answers 66 views Given that z=1+i, find the value of n\in\mathbb{Z^+} such that z^n is real. Given that z=1+i, find the smallest value of n\in\mathbb{Z^+} such that z^n is real. I'm wondering if there's an algebraic way of solving this question, aside from the obvious trial and error ... • 315 0 votes 2 answers 34 views Elementary number theory problem involving binomial theorem [duplicate] Exercise. Let a,b be real numbers, such that a+b and ab are integers. Show that \forall n\in \Bbb N: (a^n+b^n)\in \Bbb Z. I tried to use the binomial formula: (a+b)^n=\sum_{k=0}^n{{n}\... 1 vote 1 answer 91 views Can we re-write Newton's Binomial formula as a power series in \ r\  without any problems? Newton's Generalised Binomial theorem states that if \ x\  and \ y\  are real numbers with \ \vert x \vert > \vert y \vert\ (\text{note that } \left\vert \frac{y}{x} \right\vert < 1),\  ... • 12.2k 0 votes 0 answers 36 views Evaluating  \binom{n}{0} + 2 \binom{n}{1} + 2^2 \binom{n}{2} + \dots + 2^k \binom{n}{k} + \dots + 2^n \binom{n}{n}  [duplicate] With n a positive integer, evaluate the sum$$ \binom{n}{0} + 2 \binom{n}{1} + 2^2 \binom{n}{2} + \dots + 2^k \binom{n}{k} + \dots + 2^n \binom{n}{n} $$I'm pretty sure that this has to do with the ... • 315 0 votes 1 answer 33 views Is the series expansion of \frac1{(1+x)^n} same as \frac1{(1-x)^n} with (-1)^r The series expansion for$$\frac1{(1-x)^n} = \sum_{r=0}^{r=\infty}C_r^{|n|+r-1}x^r$$Is the expansion of$$\frac1{(1+x)^n} = \sum_{r=0}^{r=\infty}(-1)^rC_r^{|n|+r-1}x^r ?? (Where C is combination)
We have a binomial expansion $(x+y)^n$ , where $x,y,n$ are all some real numbers and are unknown. (n$\in$ I), (x, y ≠ 0) And we know the numerical values of $(r-2)$th, (r)th and $(r+2)$th terms of the ...