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

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1answer
25 views

Reducing Binomial Summation [on hold]

How can I reduce this summation into this $$\frac{1}{2}(1+\left(1/3\right)^{50})$$ The problem comes from the 1992 AHSME Test (problem 29)
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0answers
33 views

A generalization for Binomial Theorem, Leibniz General Rule other like functions.

$n$ & $m :=$ any value in $\{0,1,2\ldots\}$ $\Omega$ & $\beta :=$ any value in $\{1,2,3\ldots\}$ [1] If there is a function $F_\beta$ such that for some value $\Omega$ and some function $T_{...
2
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1answer
29 views

Is it possible to express the inverse of a polynomial as a series?

Is it possible to express the multiplicative inverse of a nth order polynomial i.e. \begin{equation} \frac{1}{\left[\sum_{i=0}^na_ix^i\right]^2} \end{equation} as a series using binomial theorem or ...
2
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2answers
41 views

Coefficient of $x^{103}$ in the following multinomial expansion.

What is the coefficient of $x^{103}$ in the expansion of $$(1+x+x^2+x^3+x^4)^{199}(x-1)^{201}$$ ?. The answer is an integer between $0-9$. So I wrote the given expression as $(x^5-1)^{199}(x-1)^{2}$. ...
2
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0answers
46 views

Series expansion of elliptic integral involving n th order polynomial in the denominator

My goal is to find an expansion in powers of 1/ρ of the integral: \begin{equation}I_n(\rho)=\int_\rho^{+\infty}\frac{dt}{(E_n(t))^2\sqrt{t^2-h_2^2}\sqrt{t^2-h_3^2}},\quad \rho \ge h_2\end{equation} ...
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2answers
38 views

Binomial theorem question. Find the value of the constant $k$

$$\left[(k+x)\left(2-\frac{x}{2}\right)\right]^6$$ where the coefficient of $x^{2}$ is $84$.Find the value of the constant $k$. I tried to expand the equation but got a equation of degree 6 for some ...
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1answer
6 views

Show that $x(1- (\frac{R_1-R_2}{x})^{2})^{0.5}$ = x – $\frac{(r1 – r2)^2}{2x}$

Show that $x(1- (\frac{R_1-R_2}{x})^{2})^{0.5}$ = x [1 – {0.5 $\frac{(r1 – r2)}{x}$ + … ] = x – $\frac{(r1 – r2)^2}{2x}$ by using binomial theorem it was mentioned that binomial theorem is $(x+y)^2$ ...
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2answers
43 views

Proving $(w-1)^m$ is purely imaginary.

I'm having trouble trying to prove this: Let $ m\in \mathbb Z$, m even and $w\in\mathbb C$ a primitive $2m$-th root of unity. Prove that $(w-1)^m$ is purely imaginary. What I've tried to do so ...
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1answer
23 views

Probability of Getting a Yahtzee of Fives Given Two Fives

(The following problem is from MAML, Meet 3, Round 1, December 2012, Problem 3.) In the game of Yahtzee one has a chance to get Yahtzee (5 of the same kind, such as 5 sixes) in the throw of 5 ...
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3answers
109 views

Finding the coefficient of $x^7$ in the expansion $(1 + x)^{23}$

By definition, the Binomial Theorem states: $$(x+y)^n = {n\choose 0}x^n + {n\choose 1}x^{n-1}y + {n\choose 2}x^{n-2}y^2 + \cdots + {n \choose {n-1}}xy^{n-1} + {n \choose n}y^n$$ For any $x,y\in\...
3
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1answer
52 views

On the proof of Lucas' theorem

Lucas theorem states that Let $m,n$ be two natural numbers, $p$ be a prime. Suppose that $m, n$ admit the following base $p$ representation $$m=m_0+m_1p+\cdots+m_sp^s,\qquad n=n_0+n_1p+\cdots+n_sp^...
3
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1answer
30 views

Complex Numbers and Binomial Expansion

I was able to show the above by equating the solution of $cos \ 5\theta=0$ (which gives $\pi/10$) and the solution of ${16cos}^5\theta - {20cos}^3\theta+5cos\theta = 0$ (which is what you get when you ...
9
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1answer
120 views

Intuitive ways to get formula of binomial-like sum

Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural ...
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5answers
91 views

The coefficient of $x^{100}$ in the expansion of $(1-x)^{-3}$.

Please solve it and tell me the technique so that I can solve it in examination in multiple choice questions.
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2answers
39 views

Total no of terms in the expansion of $(x+z)^{50} +(x-z)^{50}$

Please help me in solving the question and also tell me the technique so that I can easily solve it when similar question come in the form of Multiple choice questions MCQS.
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0answers
15 views

Prove that the binomial theorem for complex $r$ reduces to the integer $r$ formula

According to wikipedia the binomial theorem states that if $|x|>|y|$ and $r$ complex $$ (x+y)^r=\sum_{k=0}^{\infty}\frac{(r)_k}{k!}x^{r-k}y^k $$ where is the $$ (r)_k=\frac{\Gamma(r+1)}{\Gamma(r-...
1
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1answer
43 views

Inclusion-Exclusion principle and finite product identity

On this page, there is a proof that uses the inclusion-exclusion principle to provide a formula for the value of Euler's totient function. I would be grateful if someone could explain the reasoning ...
0
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1answer
45 views

Sum reminiscent of $(1+x)^N$ (binomial theorem)

I stumbled upon this sum while working on my thesis: $$\sum_{k=0}^N \binom{N}{2k} x^k$$ I know that $$\sum_{k=0}^N \binom{N}{k} x^k = (1+x)^N$$ But when it comes to the sum above I'm lost. Is ...
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1answer
38 views

Can we apply binomial theorem for $\quad(a+b)^\ell\quad$ if $\ell\;$ irrational.

Let be$\quad a,b\;\in\mathbb R\quad, \ell\;\in\mathbb {(R\backslash Q)} \quad $ ($\ell:$irrational) Can we apply binomial theorem for $\quad(a+b)^\ell$
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1answer
23 views

How to expand an expression containing $x^2$ by binomial theorem

I understand that the coefficient of, say $x^8$, in the expansion of $(1+x)^{10}$, would be ${10 \choose 8}$, but what about an expression like $(1+x^2)^{10}$? Would I have to square root the ${10 \...
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1answer
26 views

Problem to prove for all even integers

Prove that $5^n = 3^n + \dfrac{16n (3^{n-2})}{2} + \dfrac{256n (n-2) 3^{n-4}}{8}+ ....+ 4^n$ for all even integers. I tried finding a pattern, but was unable to do so. Any help would be ...
3
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1answer
81 views

Find $\sum_{k=1}^n \binom{2n-k}{n}(-1)^k$

Is there closed form for $\sum_{k=1}^n \binom{2n-k}{n}(-1)^k$? I got above expression for a counting exercise. I wonder that it might have the closed form but I am not sure yet. Can anyone have any ...
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3answers
72 views

Proof that combinations are equal to coefficients in the binomial expansion

Let $n\in N$, $k\in Z$, $o\leq k \leq n$. Define $C^{n}_k$ as the coefficient of $x^{n-k}y^k$ in the expansion of $(x+y)^n$ $$(x+y)^n= \sum^{n}_{k=0} C^{n}_k x^{n-k}y^k$$ Prove that ${C^{n}_{k}}={...
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1answer
37 views

Binomial expansion of $(1 - w z^{-1})^{-1}$ being $\sum_{l=0}^{\infty} w^l z^{-l}$

I am reading "Neural Networks and Learning Machines", Third Edition, Simon Haykin and at the beginning I came across at something that perplexes me. I'll quote a whole page because some contexts may ...
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2answers
76 views

Find $\sum\limits_{r=0}^n(-1)^r\binom{n}{r}^{-1}$ for $n$ even

If $n$ is an even natural number, then find $$\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$$ I tried to solve the question using conventional method, by trying to use calculus, but I ...
2
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1answer
46 views

Why does the binomial theorem for negative numbers never terminate

Why does $(a+bx)^n$ never terminate if you use negative numbers or fractions for n? Surely $(a+bx)^{-2}=\frac{1}{a^2+2abx+b^2x^2}$ and not an infinite series? Thanks
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1answer
31 views

Finding coefficient of $x^n$ in this series

I was doing an assignment question, I came across this: I understand everything (I know the theorem used in the answer), but I don't get how solution switched from first "i.e." to second "i.e.". I ...
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1answer
35 views

Find the range of values of $x$ for which the binomial expansion of $\frac{3x-1}{(1-x)(2-3x)}$ is valid

Find the range of values of $x$ for which the binomial expansion below is valid. $$\frac{3x-1}{(1-x)(2-3x)}$$ Initially I read this and thought, "ok so what can't $x$ be". You can't square root a ...
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2answers
45 views

Binomial theorem estimate for very large samples

I have around $2^{105}$ balls, of which 1 in 20 is white. I expect that when I draw a random sampling of them, roughly 5% of all balls drawn would be white. What is the probability that, if I draw $...
3
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3answers
84 views

Best way to expand $(2+x-x^2)^6$

I've completed part $(a)$ and gotten: $64+192y+240y^2+160y^3+...$ Using intuition I substituted $x-x^2$ for $y$ and started listing the values for : $y, y^2 $ and $y^3,$ in terms of $x$. $y=(x-x^2)...
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3answers
47 views

Find $k$ if given the constant term of a binomial expression?

Consider the expansion of $x^2(3x^2+\frac{k}{x})^8$. The constant term is $16,128$. Find $k$. This is simply an example of a type of question I cannot understand how to do. I have many questions: 1) ...
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6answers
713 views

Solve summation expression

For a probability problem, I ended up with the following expression $$\sum_{k=0}^nk\ \binom{n}{k}\left(\frac{2}{3}\right)^{n-k}\left(\frac{1}{3}\right)^k$$ Using Mathematica I've found that the result ...
0
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4answers
81 views

Prove that there is no term independent of $x$ in the binomial expansion of $\left(x-\frac 1x\right)^{19}$

I am dealing with a fairly simple question but I'm struggling a bit to come up with a formal demonstration on why the binomial expansion of $\left(x-\frac 1x\right)^{19}$ doesn't have a term "...
0
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1answer
43 views

Stuck on finding the term independent of $x$

For the binomial expression $$\left[x^3-\frac{2}{x^2}\right]^6$$ I calculated the general term to be $$_6C_r x^{18-5r}(-2)^r$$ So for the term independent of $x$ , the exponent $18-5r=0$ but that ...
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0answers
31 views

T0 show an equation by using binomial theorem

$$\left(1+\frac{a}{n}\right)^{(n-k)} = e^a \left(1-\frac{a(a+k)}{2n}\right)+o\left(\frac{1}{n}\right)$$ as $n\to\infty$. How the binomial theorem show this above equality? Thank you for your help!
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0answers
187 views

What is the probability of a biased coin flipping heads (probability of heads is $\frac 35$) exactly $65$ times in $100$ trials?

A biased coin flips head with a probability $\frac 35$ and tails with probability $\frac 25$. The coin is flipped $100$ times. What is the probability that heads is flipped exactly $6$5 times? I used ...
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3answers
79 views

Let $(\sqrt{3} + \sqrt{2})^5 = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$ Find $a+b$.

Let $$(\sqrt{3} + \sqrt{2})^{\color{red}{5}} = a\sqrt{3} + b\sqrt{2}, a,b \in \mathbb Z$$ Find $a+b$. I don't know if that's supposed to be $\color{red}{5}$ or $\color{red}{3}$. By binomial ...
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2answers
41 views

Application of power series/ binomial theorem in inverse sampling

I have posted this already in other forums. Apologies for cross posting. In order to establish some properties of inverse sampling, Haldane (1945) uses power series and the binomial theorem I assume....
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1answer
45 views

Doubt in a derivation

I am reading Hardy's book in divergent series and a step is making me question my intelligence. Page 178, equation (8.2.2), we have $\sum_{n=0}^{\infty}a_nx^{n+1}$ is a sequence converging to $f(x)$ ...
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3answers
61 views

Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$

Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$. I know he Binomial Theorem will be useful here, but I am struggling to use it with any certainty.
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1answer
65 views

Upper bound on $(1 + x)^n$

I'm looking for a useful upper bound on $(1 + x)^n$ in terms of $n$ and $x$. You can assume $x > 0$. Does anyone know one? An asymptotic upper bound would also be helpful.
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0answers
33 views

Under what conditions can I split a power of a binomial sum into two products?

I was reading a paper and came across a section that claimed that if $y \in \mathbb{N}$, and if $x \in [0,1]$, then for the expression: $$ \left(\frac{1}{2}+ \frac{x}{4}\right)^y $$ there exists a $...
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2answers
70 views

Finding the limit of lim${_{n \rightarrow \infty}}\left( \dfrac{n^3}{2^n} \right)$

For a class of mine we were given a midterm review; however, I just cannot figure out how to finish this one: Finding the limit of lim${_{n \rightarrow \infty}}\left( \dfrac{n^3}{2^n} \right)$ My ...
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0answers
27 views

Tricky divergent binomial expansions?

The binomial expansion of $(a+b)^n$, where $n\notin\mathbb{N}$, is given as $$(a+b)^n=a^n+\frac{n}{1!}a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+\cdots$$ In some situations, we can find the result of a ...
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1answer
76 views

Simple Question on Binomial theorems… [closed]

I have tried to solve this question by putting the value of each coefficients but it is really becoming very lengthy.... what i got was this number 10510100501... But how to get this in the required ...
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2answers
35 views

binomial coefficient where k > n

For solving binomial coefficients we have use from formula $\frac{n!}{k!(n-k)!}$ This formula only works if n > k. What happens if n < k? Is there another formula we need to use?
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1answer
30 views

Binomial expansion extension to negative powers

I know that: $$\sum_{k=0}^n {n \choose k}a^{n-k}b^k = (a+b)^n$$ But how is this extended to negative powers, for example, I came across the following line of maths, which I struggle to understand: $...
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1answer
62 views

Show that the sum $\sum_{k = 0}^n 2^k \binom{n}{k}$ is equal to $3^n$

How can I show that the sum $$ \sum_{k = 0}^n 2^k \binom{n}{k}$$ is equal to $3^n$?
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2answers
44 views

Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that $f(x)...
6
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0answers
68 views

Intuitive explanation of Extended binomial coefficient

We all are familiar with the following formula - $$\dbinom{n}r = \dfrac{n!}{(n-r)! \space r!} \space\space \space ; \space \space n>r$$ This is the binomial formula where $n$ and $r$ are ...