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Questions tagged [binomial-theorem]

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

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Confused on deciding whether its Negative Binomial or Binomial and Negative Hypergeometric or Geometric

I am currently struggling on deciding whether a question is a Negative binomial or a Binomial, and Negative Hypergeometric or a Hypergeometric SRV (Special Random Variable), As I seem to always ...
3
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2answers
48 views

Evaluate:$\frac{1}{2^{101}}\sum_{k=1}^{51} \sum_{r=0}^{k-1}\binom{51}{k}\binom{50}{r}$

Evaluate:$$\frac{1}{2^{101}}\sum_{k=1}^{51} \sum_{r=0}^{k-1}\binom{51}{k}\binom{50}{r}$$ My Attempt: I did try writing the series $(1+x)^{50}$and $(1+x)^{50}$ separately,then multiplied but could not ...
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1answer
33 views

How would I find an element or elements that has/have $x^5$ in trinomal development $(x+{1/x^{3}}+x^2)^8$ [on hold]

This is what i did. $$\left(x+{\frac{1}{x^{3}}}+x^2\right)^8 =\left(\frac{(x^{4}+1+x^{5})}{x^3}\right)^8=\frac{1}{x^{24}}(x^4+1+x^5)^8$$
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2answers
37 views

Using binomial Theorem how we can show $\frac{(x+y)!}{x!y!}\leq \frac{(x+y)^{x+y}}{x^xy^y}$?

Using binomial Theorem prove that $$\frac{(x+y)!}{x!y!}\leq \frac{(x+y)^{x+y}}{x^xy^y}.$$ I tried it as follows: It is clear that $x\leq x+y, \forall x,y\in \mathbb{N}$. Thus, by Binomial Theorem, ...
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1answer
18 views

How to find the greatest absolute term in a binomial expansion

I know that the ratio of consecutive terms should be ≥1 and I'm able to solve using that approach but I was wondering whether we could derive a general formula and how would it work?
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2answers
21 views

Question on word combinations with exclusivity

"How many 4 letter words on the alphabet {a,b,c} in which 'a' occurs exactly twice are there?" My answer is incorrect as I answered 3*3*2*2 4 letter words. However, this doesn't necessarily remove '...
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2answers
28 views

Proof $\mid(a - b)^{\frac{1}{n}}\mid \leqslant \mid a^{\frac{1}{n}} - b^{\frac{1}{n}}\mid$

We have $(a + b)^{n} \geq a^{n} + b^{n}$ Proof: $\mid a - b \mid ^{\frac{1}{n}} \geq \mid \mid a \mid^{\frac{1}{n}} - \mid b\mid^{\frac{1}{n}}\mid$
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0answers
16 views

analytic integration for binomial term?

I have questions about method of analytic integration I have an expression $$p(N|a,s,b) = \frac{\exp[(-as+b)](as+b)^N}{N!} * \frac{\exp(-gs)(gs)^A}{\Gamma(A+1)}*\frac{\exp(-hb)(hb)^B}{\Gamma(AB+1)}$...
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4answers
68 views

What is the coefficient of $x^{11}$ in $(3x-9)^{19}$?

I am currently studying for finals, and I do not know how to do this problem from my study guide. I have tried to watch a few YouTube videos and I know that I will end up with $3x^{11} \times (-9)^8$, ...
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1answer
16 views

Let n belongs to +ve integer and $(1+x+x^2)^n=\sum_{r=0}^{2n} {a_rx^r}$ prove that: $a_r=a_{0<r<2n}$

Let n belongs to +ve integer and $$(1+x+x^2)^n=\sum_{r=0}^{2n} {a_rx^r}$$ prove that: $$a_r=a_{2n-1},{0<r<2n}$$ as well as prove that $$\sum_{r=0}^{ n-1} a_r=\frac{1}{2}(3^n-a_n)$$. I tried to ...
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1answer
31 views

Complex number identity question

If $n$ is even and $z=\cos\theta+i\sin\theta$. By expressing $z^n$ in two ways, show that $$\binom{n}{0}-\binom{n}{2}+\binom{n}{4}-\cdots+(-1)^{\frac{n}{2}} \binom{n}{n}=2^{\frac{n}{2}}\cos\left(\...
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3answers
30 views

How to expand $x^{1/3}-c^{1/3}$ into $(x-c)y$ for some $y$

How to expand $x^{1/3}-c^{1/3}$ into $(x-c)y$ for some $y$ I know $x^3-c^3=(x-c)(x^2+xc+c^2)$ but I can't figure out how to pull this off with $1/3$ instead of $3$
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2answers
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A proving question based on binomial theorem [closed]

$$C_0-C1(a-1)(b-1)(c-1)_+C_2(a-2)(b-2)(c-2)+.... (-1)^nC_n(a-n)(b-n)(c-n) $$=0 I tried to solve this problem by using multinomial theorem but was not able to proceed further please help me out.
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1answer
47 views

Question based on binomial theorem and complex numbers. [closed]

Prove that $$C_1+C_5+ C_9+... =\frac{1}{2}\bigg(2^{n-1}+2^{n/2}\sin\frac{n\pi}{4}\bigg)$$ Here $C_i$ denotes the binomial coefficient $\binom ni$. I tried to solve this problem by using de Moivre's ...
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2answers
54 views

One term of (2π+5)^n = 288000π^8, what's n?

Without using calculator what's the value of n? Using binomial expansion I get: nCp * 2n-p * πn-p * 5p = 288000π8 Easily I know that n-p=8, by the π's power Then the power of 2 is also 8, so I can ...
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2answers
75 views

Weighted sum of product of binomial coefficients

I am trying to evaluate the sum $\displaystyle \sum_{n=1}^N \sum_{k=1}^n k\binom{n}{k} \binom{N-n}{k}x^k$, Here $x$ is some positive real My approach so far has been to first to compute the ...
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0answers
253 views

Probability question: calculating probability of attendance in training camp.

Assume there are $400$ athletes in a training camp, who are required to attend the morning drill starting at $4$ am. The attendance in morning drills is $70\%$, i.e. on an average, $280$ athletes are ...
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5answers
63 views

How to show that $2^n > n$ without induction

I'm solving exercises about Pascal's triangle and Binomial theorem, and this problem showed up, however I don't have any clue on how to solve it The sum of ${n\choose p}$ from $p=0$ to $n$ is the ...
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1answer
53 views

Proving $\int ^{2\pi} _{0} (\cos(t)^{2n})={2n \choose {n}}\frac{2\pi}{2^{2n}}$

using the following result $\int _{\gamma}(z+ \frac{1}{z})^{2n}\frac{dz}{z}= {2n \choose {n}}2\pi i $ Prove $\int ^{2\pi} _{0} (\cos(t)^{2n})={2n \choose {n}}\frac{2\pi}{2^{2n}}$ I cant see how ...
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4answers
65 views

I tried this question by using pi denoting method and got a big equation what to do

Find coefficient of $x^6$ in $(1+x)(1+x^2)^2.....(1+x^n)^n$
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1answer
28 views

How does it follow from the Pascal's Triangle that binomial coefficient are integers

So I was reading this lemma which states: Let $m,n$ be natural numbers such that $1 \leq m \leq n$. Then \begin{equation*} {n\choose m-1} + {n\choose m} = {n+1\choose m}. \end{equation*} It follows ...
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0answers
16 views

Analytic integration of profile likelihood vs marginal likelihood?

I have questions about 1. Method of analytic integration 2. Statistics (marginal vs profile) I have an expression $p(N|a,s,b) = \frac{exp[(-as+b)](as+b)^N}{N!} * \frac{exp(-gs)(gs)^A}{\Gamma(A+1)}*\...
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0answers
32 views

Use the Binomial Formula to show that

There's the problem $$ \binom{n}{0}+\binom{n}{2}+\cdots=\binom{n}{1}+\binom{n}{3}+\cdots=2^{n-1} $$ I know how to do operations with the binomial formula but I do not understand what that set up is ...
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2answers
125 views

Strict Bernstein Inequality

We define the entropy function by $$H(x)=x\ln\frac{1}{x}+(1-x)\ln\frac{1}{1-x} \quad \text{ for } 0 \leq x \leq 1 $$ where $H(0)=H(1)=0$. a) For integers $0\leq k\leq n$ prove that $$\...
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91 views

Strengthening of Bernstein Inequality

Problem. Let $X_1, . . . , X_n$ be independent random variables such that $\mathbf{E}(X_k) = 0$ and $|X_k| \leq 1$ for $k = 1, . . . , n$. Let $X = X_1 + . . . + X_n$. Prove the following ...
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1answer
31 views

Binomial Expansion rth term

Given that there's no term on $x^3$ in $$(k + 2x) (1 - 3/2 x) ^6.$$ Evaluate the value of $k$ Difficult After expanding $(1 - 3/2 x)^6$ up to the 3rd term I get $1 - 9x + \frac{135}{2} x^2 - \frac{...
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1answer
57 views

Showing that $\left(1+\frac{\varepsilon^2}{17n}\right)^n-1\leq \frac{\varepsilon^2}{16}$ when $0\leq \varepsilon\leq 1$

Question Show that $\left(1+\frac{\varepsilon^2}{17n}\right)^n-1\leq \frac{\varepsilon^2}{16}$ when $0\leq \varepsilon\leq 1$. This inequality appeared in the middle of an argument I was reading and ...
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0answers
70 views

Proof of Newton's Generalized Binomial Theorem (without Calculus)

I'm writing an article for derivates, I've already prooved Newton's Binomial Theorem, but I want to proof that the expresion $$(a+b)^r=\sum_{i=0}^\infty\binom{r}{i}a^ib^{r-i}$$ works for all $a,b,r\in\...
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1answer
39 views

How to prove Binomial Theorem

I have got through part a) and b) The answer is 365 and 364, respectively , however I'm not able to tackle part c)
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1answer
39 views

Evaluate $lim_{n \to \infty} (n + 1)^{\frac{2}{3}} - (n)^{\frac{2}{3}}$ using the generalized binomial theorem

I have already checked this question and have understood the approaches using the Product Rule for Limits, and the one using the Mean Value Theorem. I was wondering whether it is possible to evaluate ...
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2answers
43 views

How can I find the number of dissimilar or distinct terms in $(1+x + x^2 +x^3)^n$?

How can I find the number of dissimilar or distinct terms in $(1+x + x^2 +x^3)^n$ ? I know it would be $\binom{n+r-1}{r-1}$ when they are $a , b, c ,d$ instead of $1 , x , x^2 , x^3$. But how ...
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2answers
35 views

How to find the sum of the given series: $\sum_{k=0}^{\min(n ,m)} \binom{n}{k} \binom{m}{k}$?

I wonder whether it is possible to calculate the following sum that involves the Binomial coefficients $\sum_{k=0}^{\min(n ,m)} \binom{n}{k} \binom{m}{k}$ .
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2answers
88 views

If $y=\frac25+\frac{1\cdot3}{2!}\left(\frac25\right)^2 +\frac{1\cdot3\cdot5}{3!}\left(\frac25\right)^3+\ldots$ then what is the value of $y^2+2y$?

If $$y=\frac25+\frac{1\cdot3}{2!}\left(\frac25\right)^2 +\frac{1\cdot3\cdot5}{3!}\left(\frac25\right)^3+\ldots$$ then what is the value of $y^2+2y$? This is a question from my coaching material in ...
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2answers
59 views

How is $C_0^nC_n^{n+1}+ C_1^nC_{n-1}^{n}+C_2^nC_{n-2}^{n-1}+\dots+C_n^n C_0^{1} =2^{n-1}(n+2) ?$ [closed]

How is $$C_0^nC_n^{n+1}+ C_1^nC_{n-1}^{n}+C_2^nC_{n-2}^{n-1}+\dots+C_n^n C_0^{1} =2^n(n+2) ?$$ I have no idea how to approach this problem. There is no solution given in my book. There doesn't seem ...
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1answer
29 views

Small induction problem.

Show with the help of induction for all natural numbers n, where a and b are real numbers and: $a \neq b$ and $ a + b > 0$ $$(a + b)^n ≤ 2^{n−1}\cdot(a^n + b^n) $$
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1answer
30 views

Trying to evaluate the constant term [closed]

Suppose you are given that $$\biggr (\sqrt[3]x - \dfrac{1}{\sqrt x} \biggr )^{15}$$ I'm trying to evaluate the constant term. $$\sum^{15}_{n = 0} \binom{6}{r}x^{5-r}\cdot -\dfrac{\sqrt x}{x}$$ ...
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2answers
69 views

Calculating coefficient

I have a generating function, $$ \frac{(1-x^7)^6}{(1-x)^6} $$ and I want to calculate the coefficient of $x^{26}$ Solution for this is, $$ {26+5 \choose 5} - 6{19+5 \choose 5} + 15{12+5 \choose 5} - ...
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2answers
84 views

Numerical differentiation with Binomial Theorem

In George Shilov's Elementary Real and Complex Analysis, there is a problem which asks us prove If $f$ is twice differentiable on some open interval and the second derivative is continuous at $x$, ...
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3answers
51 views

Find the coefficient of $x^3$ in the expansion of $(1+x+x^2)^3$ using Binomial theorem. [closed]

Find the coefficient of $x^3$ in the expansion of $(1+x+x^2)^3$ using Binomial theorem.
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4answers
39 views

Intuitive understanding of the binomial theorem?

I'm trying to understand the thought process, of how I might come upon the binomial theorem intuitively by thinking about combinatorics, can someone help?
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3answers
45 views

Coefficient in binomial expansion [closed]

Whats is the coefficient in front of $x^{50}$ in the binomial expansion $$(1+x)^{1000} + (1 + x)^{999}\cdot x^1 + ... + (1+ x)^1\cdot x^{999} + x^{1000}$$
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1answer
25 views

2n-th derivation of Legendre Polynomial

Let $u_n(x)=(x^2-1)^n$ Show that $\frac{(d^{2n}u_n(x)} {dx^{2n}} = 2n!$ $(x^2-1)^n = (x-1)^n(x+1)^n $ and then a should use Leibnitz formula. I thought if I write Leibnitz formula as a binomial I ...
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2answers
90 views

Binomial sum of $\sum\limits_{k=0}^{n} k \binom{n}{k} (-1)^{k-1} 2^{n-k}$

Evaluate $\displaystyle{\sum_{k=0}^{n} k \binom{n}{k} (-1)^{k-1} 2^{n-k}}$ So if the $k$ wasn't here this binomial sum would be really easy to solve. How would you evaluate this then?
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3answers
30 views

What is the coefficient of the term $x^3y^5$, as a result of the binomial expansion of the following term?

We have the term $(1+xy+y^2)^n$ If we expand it using the binomial theorem, why is the factor of the term $x^3y^5$ the following: $4{n\choose 4}$? (The binomial coefficient n choose 4 multiplied by 4)...
3
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2answers
42 views

Binomial Expansion to the power of a non-natural number

What is the reasoning that $|x|$ has to be less than $1$ for $(1+x)^n$ when $n$ is not a natural number?
4
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1answer
63 views

Combinatorial Analysis - Specific problem

I am having difficulty modeling a combinatorial analysis on a particular problem, I wanted to isolate some generic form to count how many valid arrangements exist in a given problem, can anyone help ...
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2answers
50 views

Prove that $(n+2)^n < (n+1)^{n+1}$ for all $n \in \Bbb N$

Is there a simple(ish) way of proving this? I got to this step when I was trying to show that the sequence $a_n = (1 + \frac{1}{n})^n$ is increasing for all $n \in \Bbb N$. It came up after I expanded ...
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3answers
31 views

Find the values of $a$ and $b$ when the binomial expansion of $\frac{1}{(1+ax)^b}+\frac{1}{(1+bx)^a} = 2-6x+15x^2$

Find the values of $a$ and $b$ when the binomial expansion of $$\frac{1}{(1+ax)^b}+\frac{1}{(1+bx)^a} = 2-6x+15x^2$$ So I set up the two equations: $$(-b)(ax)+(-a)(bx)=-6x$$ and $$\frac{(-b)(-b-1)(ax)...
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votes
2answers
41 views

Binomial expansion of $\frac{1}{1+x+x^2}$ up to the first three terms [closed]

Binomial expansion of $\frac{1}{1+x+x^2}$ up to the first three terms I am unsure where to start with this as i cannot put it into partial fractions, so don't really have an idea on where to start and ...
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3answers
85 views

Finding the term independent of $x$ from $2$ expansions.

I was asked to find the first $3$ terms of the expansion $\left(3-\frac1{9x}\right)^5$ and was further asked to find the term independent of x in the expansion of $\left(3-\frac1{9x}\right)^5(2+9x)^2$....