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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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59 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
40 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
28 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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2answers
35 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
19 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 $\left(3-\frac1{9x}\right)^5(2+9x)^2$....
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1answer
26 views

Find the binomial expansion of the function up to the first 3 non-zero terms $\sqrt[3]\frac{1+2x}{1-x}$

Find the binomial expansion of the function up to the first 3 non-zero terms $$\sqrt[3]\frac{1+2x}{1-x}$$ The function can be broken into $(1+2x)^{\frac{1}{3}}$ and $(1-x)^{\frac{-1}{3}}$ where $(1+...
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1answer
18 views

Explanation In the orderings involving $2$ kinds of elements

A population of $a$ indistinguishable alphas and $b$ indistinguishable betas can be arranged in $\binom{a+b}{a}=\binom{a+b}{b}$ distingishable orders. Any permutations among the alphas, or among ...
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2answers
49 views

Rule of 72 binomial

Rule of 72? So you have to work out the time it takes for an investment to double. With interest rate = x So I get: (1+x)^n = 2 Then I get how the 72 rule is formed through solving for n, giving n ...
2
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1answer
39 views

Probability and Statistics: Over-booking flights

I am finding difficulty trying to answer the second part of this question. The answer is one this website but I do now get how they worked it out. https://nrich.maths.org/4932 I also know that ...
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2answers
26 views

Using the Binomial Theorem to expand the magnitude of the difference of two vectors

I have the following expression that I need to expand using the Binomial Theorem: $$\frac{1}{\mid\vec{r}-\vec{d}\mid}$$ Now the Binomial Theorem is the following: $$(x+y)^r = \sum^{\infty}_{k=0}\binom{...
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3answers
54 views

prove that ${2n \choose n} \le 4^{n}$

i need to prove that ${2n \choose n} \le 4^{n}$ for all $ n \in N $. I tried to prove it that way: $$ 4^{n} = (1+1)^{2n} = \sum_{i=0}^{2n} {2n \choose i}1^{2n-i}1^{i} = \sum_{i=0}^{2n} {2n \choose i} ...
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4answers
58 views

How to prove divisibility of a number using the binomial expansion? [closed]

I have the following problem: Prove that $6^n-1$ is always divisible by 5 using the binomial expansion of $(5+1)^n$. How can I do this? I don't know how to begin, as I don't see how the binomial ...
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2answers
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Finding the value of a double summation [closed]

The question says to find the value of $$\sum ^{n} _ {r=1} \sum ^ {r-1} _ {p=0} \binom{n}{r} \cdot \binom {r}{p} \cdot 2^p$$ I have no idea how to deal with the double summation, I just know that ...
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1answer
78 views

Prove that $(x+y)^p=x^p+y^p \mod p$ where $p$ is a prime.

Let $p$ be a prime, we want to prove that $(x+y)^p=x^p+y^p \mod p$. The left handside looks like something we can throw the binomial theorem at, also in the previous proof the book asked to prove that ...
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1answer
28 views

Bacteria that doubles every time step with a probability of p or dies. What is expected number of bacteria after n time steps?

As stated in the problem. Given a bacterium that at every time step $t$, either divides with a probability $p$ or dies off. What is the expected number of bacteria after $n$ time steps? Does this use ...
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2answers
55 views

Prove $3^{n−2} n (n − 1) = \sum_{k=2}^{n} \binom{n}{k} (k)(k − 1)2^{k−2}$ for $n\ge3$

The question asks for a combinatorial proof only. In my attempt, I rewrote $3^{n-2}$ as $(1+2)^{n-2}$. Then using binomial theorem, i was able to get $(1+2)^{n-2} = \sum_{k=2}^{n} \binom{n}{k} 2^k$....
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1answer
38 views

Power Series / Taylor Expansion of $\frac{1}{\left(1-t^2\right)^{\frac{1}{2}}}$

The question: $\left(\frac{1+t}{1-t}\right)^{\frac{1}{2}}\:=\:\sum _{n=0}^{\infty }\:a_nt^n$ The question asks for what is $a_n$ These are the steps I've done: $\left(\frac{1+t}{1-t}\right)^{\frac{...
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0answers
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Characteristic Function and Generalized Binomial Theorem

could you please help me on the following problem? I have the characteristic functions of two random variables $X$ and $Y$, denoted $\phi_X(u)$ and $\phi_Y(u)$, such that: \begin{equation*} \...
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1answer
65 views

Approximate $\sqrt{7}$ using binomial theorem [duplicate]

How does one deduce the approximation of $\sqrt{7}$ to be $\frac{10837}{4096}$ by taking $x = \frac{1}{64}$ in the expansion of $\sqrt{1-x}$? How should you approach such a question? I assume the ...
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0answers
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Binomial expansion; first four terms when $x < a$ or $x > a$

Using the Binomial Theorem, how do you find the first four terms of the expansion of$$(x+3)^{-4}$$when $x<3$ and when $x>3$ ?
2
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1answer
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O level Expansion problem. [duplicate]

yep, title sucks.. Q: The expansion of (1+px+qx2)8 = 1+8x+52x2+kx3. Find the values of p, q and k. I found the values of $p$ and $q$ and they are: p = 1 q = 3 But I am unable to find the value ...
2
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1answer
38 views

Prove -1 Transformation (A binomial identity):$\binom{-p}{q}=\binom{p+q-1}{q}(-1)^q$

Prove the following binomial identity : $$\begin{align*} \binom{-p}{q}=\binom{p+q-1}{q}(-1)^q \end{align*} $$ I played with $\binom{-p}{q}$ for a while but found nothing .
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1answer
33 views

Greatest term of this expansion?

If in the expansion of $(x+a)^{15}$ the $11^{\text{th}}$ term is G. M. i.e. geometric mean of $8^{\text{th}}$ and $12^{\text{th}}$ terms then which term is the greatest term among the terms in ...
4
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2answers
73 views

Find the number of 5-member committees which include at least two Republicans

There are 10 Republicans, 8 Democrats and 2 Independent legislators eligible for committee membership. How many 5-member committees exist which include at least two Republicans? My Work : $C(20,5)...
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2answers
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If there are m items of one kind,n items of another kind and so on, then the number of ways of choosing r items of these items is

Please prove this result: If there are m items of one kind,n items of another kind and so on, then the number of ways of choosing r items of these items is =Coefficient of $x^r$ in $(1+x+x^2+...+x^...
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0answers
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The number of ways in which n identical items can be divided into r groups so that no group contain less than m items and more than k$(m<k)$ is

Please prove this result The number of ways in which n identical items can be divided into r groups so that no group contain less than m items and more than k$(m<k)$ is =Coefficient of $x^n$ in ...
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4answers
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I don't see how the binomial theorem relates to the principle of inclusion and exclusion?

I'm learning discrete maths as a hobby at the moment and I got stuck when the tutor starting relating the binomial theorem to the principles of inclusion and exclusion. The video I was watching is ...
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2answers
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Find the coefficient of a negative indexed $x$ in a series expansion

Find the coefficient of x-10 in the expansion: (2-1/x2)8 ANS: -448 I've tried using the General Term formula and got stuck at -x2r = x-10. Also, I tried expanding the equation but it doesn't look ...
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2answers
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Expansion of $(\frac{x+2}{x})^{-\frac{1}{2}}$ and then find an approximate value…

First, obtain the four terms in the expansion of $ (\frac{x+2}{x})^{-\frac{1}{2}}$ then let x = 100 and use the result to find an approximate value of $(\frac{450}{51})^{\frac{1}{2}}$. I am stuck ...
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2answers
38 views

$\sum_{k=0}^n \binom{n}{k}x^k(1-x)^{n-k}k$? [duplicate]

I cant wrap my head around simplifying the following sum: $$\sum_{k=0}^n \binom{n}{k}x^k(1-x)^{n-k}k,$$ where $0<x<1$. I tried to apply standard formulas here.
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3answers
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Induction step in proof that $\binom{s}{s} + \binom{s + 1}{s} + \cdots + \binom{n}{s} = \binom{n + 1}{s + 1}$

Prove by induction that (binomial theorem) $$ \binom{s}{s} + \binom{s+1}{s} + \dotsb + \binom{n}{s} = \binom{n+1}{s+1} $$ for all $s$ and all $n>s$. I used base case $s=0$, and I got my base case ...
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3answers
61 views

Prove that $3^{16} -33$ and $3^{15} +5$ is divisible by 4 by means of binomial theorem

This is a question that I found in a textbook: Given that $p=q+1$, $p$ and $q$ are integers, then show that $p^{2n} - 2nq-1$ is divisible by $q^2$ given that $n$ is a positive integer. By taking a ...
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2answers
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Express $\frac x{x^2-3x + 2}$ in the partial fraction form and show that its partial fraction $x^3$ can be neglected

I am having trouble solving a multi part question. Express $ \frac x{x^2-3x + 2} $ in the partial fraction form. The answer I got was $\frac2{x-2}-\frac1{x-1}$ . The problem comes when they asked: ...
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1answer
58 views

Expand $\sqrt{1-x}$ up to and including the term $x^2$ (How should i proceed further)

I am trying to solve a 2 part question, the first part is to expand $\sqrt{1-x}$ up to and including the term $x^2$ which I did. This gives me $1-(0.5)x-(0.125)x^2 + ...$ However, the 2nd part of ...
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1answer
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When proving $\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}$ per induction over $n$, why don’t we need induction over $r$?

Proving this formula by induction is done by induction over $n$: $$ \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1} $$ My question is: Why don't we need to show this seperately for $r \to r+1$ too?
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2answers
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Binomial expansion lower bound $A^n + B^n \le (A+B)^n$ for non-integer $n$

By the Binomial expansion for integer powers, $$ (A+B)^n = \sum_{k=0}^n {n\choose{k}} A^{n-k} B^{k}$$ (I'm assuming $A,B\ge 0$) and so we get the easy estimate $A^n + B^n \le (A+B)^n$ for any ...
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2answers
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Prove $\sum_{r=0}^n \binom{n}{r} \binom{n+r}{r} (-2)^r =(-1)^n\sum_{r=0}^n \binom{n}{r}^2 2^r$ [closed]

Here is a combinatorics question that I am struggling. Prove $\sum_{r=0}^n \binom{n}{r} \binom{n+r}{r} (-2)^r =(-1)^n\sum_{r=0}^n \binom{n}{r}^2 2^r$ I tried to simply the binomial coefficients on ...
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1answer
72 views

simplifying triple summations from series expansion

I have the following equation I would like to extract $x^k$ out of $$ \sum^\infty_{k=0}x^k \Gamma \bigg(l,\frac{ax-1}{a^2}\bigg) $$ I start by expanding the incomplete gamma function, and then the ...
4
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1answer
107 views

How to prove the two identities

Let $m,n$ be positive integers and $N=\{1,\ldots, n\}$. Try to prove the two following identites: For $m<n$, we have $$\sum_{A \subseteq N} (-1)^{\left| A\right|} \left(\sum_{j \in A} x_j\right)^m ...
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1answer
39 views

Induction based on combinations and binomial theorem: ${}^nC_0+{}^{n+1}C_1+{}^{n+2}C_2+\dots+{}^{n+p}C_p={}^{n+p+1}C_p$ [duplicate]

I was looking at some questions in a Cambridge text and I reached this question however I am at it for 1 hr and can't seem to get the proof right. Any help ?
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0answers
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Could someone help clarify the meaning of this pre-calculus question on Binomial Expansion?

I'm taking an online pre-calculus course and I'm currently working on a unit about Binomial expansion. I've come across an oddly worded (at least to me) question on coefficients: "What coefficient ...
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1answer
63 views

A lower bound for $(1-e^x)^n$

I want to find a lower bound for $$(1-e^x)^n$$ $n$ integer, $x$ real, and $1-e^x\geq 0$. One lower bound is (Bernoulli's inequality) $$(1-e^x)^n\geq 1-ne^x$$ But I need a tighter lower bound that is ...
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2answers
42 views

What kind of expansion is this?

In a paper I came across an expansion like this: $$\cos(m\theta) = C_m^0\cos^m(\theta) - C_m^2\cos^{m-2}(\theta)(1-\cos^2(\theta)) + C_m^4\cos^{m-4}(\theta)(1-\cos^2(\theta))^2 + ... (-1)^nC_m^{2n}\...
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0answers
25 views

Generalizing $(^4C_0)^2-(^4C_1)^2+(^4C_2)^2-(^4C_3)^2+(^4C_4)^2=\,^4C_2$

I got this question, and I have a bit of an issue with part (b). I managed to prove the attached result; however, it does not really fit the above result (as it said generalise the result and then ...
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1answer
42 views

Sum of weighted binomial coefficients [duplicate]

I am struggling with computing the following sums: $$\sum_{k=1}^{n}k\binom{n}{k}=\binom{n}{1}+2\binom{n}{2}+...+n\binom{n}{n}$$ and $$\sum_{k=0}^{n}\frac{1}{k+1}\binom{n}{k}=\binom{n}{0}+\frac{...
2
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3answers
64 views

Writing $(m+2)^n-(m-2)^n$ in summation notation.

I have expanded $(m+2)^n-(m-2)^n$ the following way: $$(m+2)^n-(m-2)^n = 2 {n \choose 1}m^{n-1}+ \dots + {n \choose n-1}m2^{n-1}-{n \choose n-1}m(-2)^{n-1}+{n \choose n}2^n - {n \choose n}(-2)^n$$ ...
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1answer
86 views

A Subtle Point in Rudin's Proof on $e$

When Rudin proves the theorem $\lim_{n\rightarrow \infty} (1+\frac{1}{n})^n=e$, he uses the following claim: Let $t_n= (1+\frac{1}{n})^n=1+1+\frac{1}{2!}(1-\frac{1}{n})+\frac{1}{3!}(1-\frac{1}{n})(...
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1answer
51 views

Polynomial of $n-1$ degree

Given a polynomial in $\mathbb{C}$, I want to show that a polynomial of the form $$P_n(z)=\sum^n_{i=0} a_iz^i$$ can be decomposed into something like $$P_n(z)=(z-z_0)\cdot Q_{n-1}(z),$$ where $Q$ is ...
2
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1answer
51 views

Which type of random walk has distribution of “scaled” binomial coefficient?

We know that, for a 1D symmetrical random walk, ${\displaystyle p = 1/2, q = 1/2}$, with equal walk step length, after n steps, its probability distribution will be proportional to binomial ...
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2answers
41 views

How to prove $n^p>\log n$ for a fixed $p\in (0,1)$ when $n\rightarrow \infty$? [duplicate]

I want to prove: $$n^p>\log n$$ for a fixed $p\in (0,1)$ when $n\rightarrow \infty$. I test by some big number and it seems true. But how to prove that?