Questions tagged [binomial-theorem]

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

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32 views

How do you find the coefficient of $x$ in $(x + 1)^2$?

I want to learn how can I find out the coefficient of the variable $x$ in the expression $(x + 1)^2$. It is a case of a perfect square expansion.
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18 views

approximated value Using Binomial theorem Expansion

Find An approximated value for 4√630 , (2.9)5 Using (i) Binomial theorem Expansion (ii) Differentiation Methods enter image description here
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2answers
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How may one solve problems over expressions like $(2+px)^6$ without the binomial theorem?

A friend of mine posed a problem on a mathematics discord server. The coefficient of the $x^2$ term in the expansion of $(2+px)^6$ is $60$. Find the value of the positive constant $p$. I ...
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1answer
32 views

How do I simplify factorials?

Given the following equation: $$\frac{n!}{(n-k)!k!}*\frac{1}{n^{k}}=\frac{1}{k!}*\frac{n*(n-1)...(n-k+1)}{n^{k}}$$ (I used * as the multiplication sign). I understand $\frac{1}{k!}$ but $$\frac{n*(...
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4answers
52 views

Evaluate the sum $\sum_{n=0}^{\infty} \frac{2n}{8^n}{{2n}\choose{n}}$

Evaluate the sum $\sum_{n=0}^{\infty} \frac{2n}{8^n}{{2n}\choose{n}}$ I am unable to find a way to solve this sum. I have never seen sums involving binomial coefficients multiplied by $n$. Help will ...
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1answer
51 views

Find the generating function and the number of integers solutions for $x_1 + x_2 + x_3 + x_4 = r$, where $-3 \leq x_i \leq 3$. [closed]

Well, we have $x_1 + x_2 + x_3 + x_4 = r$ where $x_i\in\{-3,-2,-1,0,1,2,3\}$ Then the generating function is given by $f(x)=(x^{-3}+x^{-2}+x^{-1}+x^{0}+x^1+x^2+x^3)^4 = \frac{(x^{6}+x^{5}+x^{4}+x^{3}+...
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3answers
31 views

Closed for solution for $\sum_{k = 0}^{n} Q^{k} ( 1 - Q) ^ {k}$

I know the binomial expansion formula: $$ (1 + x)^n = \sum_{k = 0}^{n} {n \choose k}x^k $$ However, I am trying to find (if there is any) a closed-form solution for the following equation. $$ \sum_{...
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0answers
39 views

calculate the given sum (related to Newton's binomial formula) [duplicate]

i was given the following question: given n is a positve integer, calculate the given sum: $$ {n \choose 1} -2{n \choose 2} +3{n \choose 3}+ \ldots +(-1)^{n-1}\cdot n {n \choose n} $$ I looked at ...
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1answer
35 views

Radius of convergence for binomial series (2)

I'm having trouble calculating the radius of convergence for for the following binomial series. More in detail, I'm having trouble finding $c_k$ and $c_{k+1}$ for the following series: $$ \sum_{k=0}^...
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37 views

Binomial expansion of $(x+1)^{-1}$

Why are the binomial expansions of $(x+1)^{-1}$ and $(1+x)^{-1}$ different? The expansion of $(1+x)^{-1}$ is - $$(x+1)^n= 1+ nx + \frac{n(n-1)}{ 2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3\cdots$$ where the ...
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2answers
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How do I solve this problem using binomial theorem with square root in it?

The question is: Find the coefficient of $\frac{1}{x\sqrt x}$ in the expansion of $(x^2 - \frac{1}{2\sqrt x})^{18}$. I have included a photo to make it easier to read because I do not know how to ...
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1answer
45 views

How can I show binomial series converges to $\sqrt {2}$?

$\sum_{n=0}^{\infty} 2n\binom{-\frac{1}{2}}{n}(-\frac{1}{2})^n = \sqrt{2}$ From wolfram alpha, it says that above series including binomial term $\binom{-\frac{1}{2}}{n}$ converges to $\sqrt{2}$. I ...
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1answer
50 views

Evaluation of a tricky binomial sum

The Question: To prove that: $\frac{3!}{2(n+3)} = \sum_{r=0}^{n}{(-1)^r\frac{\binom{n}{r}}{\binom{r+3}{r}}}$ My Attempt: I start off by writing $\sum_{r=0}^{n}{(-1)^r\frac{\binom{n}{r}}{\binom{r+3}...
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1answer
32 views

Reciprocal of square root of a binomial to series.

by square root algorithm and long division (or by binomial theorem) it is simple matter to find $1/\sqrt{(1-x^2)} = (1+x^2/2 + 3x^4/8 + 5x^6/16 + ...)$ > I am new to this kind of thing. Can someone ...
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1answer
44 views

How to show $\sum_{i=0}^{2m} (-1)^{i}\binom{2m}{i}^{2} = (-1)^m\binom{2m}{m}$ [duplicate]

I am trying to show that for any positive integer m, $$\sum_{i=0}^{2m} (-1)^{i}\binom{2m}{i}^{2} = (-1)^m\binom{2m}{m}$$ Intuitively this seems to be true, $m = 0$ both sides evaluate to $1$, $m = 1$...
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20 views

Inverse of the difference of matrices

Let $A\in R^{n\times n}$ (invertible and symmetric), $B\in R^{k\times k}$ (invertible and symmetric), $C\in R^{n\times k}$, $D\in R^{k\times n}$, $CBD$ is (invertible and symmetric). I am trying to ...
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0answers
15 views

Find the misterious function $F(r,a,q)$.

I've the pattern bellow where we can spot the binomial coefficients. Observing the pattern, for each row, it seems that it arises from $$ \left(\left(r a^{q} \right)^{\frac{1}{q}} +F(r,a,q)\right)^{n} ...
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1answer
44 views

Sum of product of binomials

While working on a combinatorics problem, I found that this result had to be true: $$\sum_{i=0}^n\frac{(a-b)^i(b-c)^{n-i}}{i!(n-i)!}=\frac{(a-c)^n}{n!}$$ for $a\geq b\geq c$, with $a,b,c\in\Bbb N$. ...
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2answers
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Series Expansion and Big-Oh Notation: Expanding $\sqrt{x^2+\mathcal{O}(x^3)}$ to get $x+\mathcal{O}(x^{3/2})$

I've been reading through a thesis and trying to rederive all of the equations used. However, I've come across an expansion that I'm unsure on. The following is a simplified form of two consecutive ...
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2answers
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Confused about the meaning behind making x = 2 in the binomial theorem

So the binomial theorem states $(1+x)^n = \sum^{n}_{k=0}$$n \choose k$$x ^ k$. Now I understand that each term of the sum represents the number of ways to arrange 1 and $x$ out of $n$ choices, so ...
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1answer
21 views

Radius of convergence for binomial series

The question is simple: determine the radius of convergence for the given series. I have done this for other problems using the following rule which our book described: If the given series is $$ \...
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1answer
128 views

Tighter upper bound on $x$ where $2^x \leq \sum_{i=0}^m{{x \choose i}\lambda^i}$

We have the following inequality: $$2^x \leq \sum_{i=0}^m{{x \choose i}\lambda^i}$$ All the variables are in $\mathbb{N}_{>0}$ I need to find a tight upper bound for $x$ using $m,\lambda$. In ...
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1answer
58 views

why can't we use binomial theorem to expand this expression

I was recently told that while expanding expressions of the form $$(A+B)^n$$ where A and B are square matrices of same order and n is a natural number then $$(A+B)^n =$$ $$ {{n}\choose{0}}A^{0}B^{...
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1answer
32 views

May I know why is the fact that P(The best is among the first n) is calculated in this way?

I am trying to solve this problem: On the basis of an interview, the N candidates for admission to a college are ranked in order according to their mathematical potential. The candidates are ...
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0answers
19 views

binomial theorem combinatorics consideration question [duplicate]

i have an identity that i need to prove in 2 ways $$\sum_{k=m}^n \binom k m = \binom {n+1} {m+1}$$ the first way was to prove using induction which wasn't a problem since n≥m so just do the base and ...
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1answer
21 views

Fractional exponent for binomial theorem

If I am trying to expand $(a+b)^{\frac{2}{3}}$, can I use the binomial theorem like so: $$\sum_{k=0}^{\frac{2}{3}}{\frac{2}{3}\choose k}a^{\frac{2}{3}-k}b^k$$ or will that not work, since the last ...
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1answer
59 views

$q$-analogue of $\sum_{k=0}^n \, {n \choose k} = 2^n $

Is there a $q$-analogue of the formula $\sum_{k=0}^n \, {n \choose k} = 2^n $ in terms of the $q$-binomial coefficient ${n \choose k}_q$ and $(2^n)_q=(1+q)...(1+q^n)$?
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1answer
40 views

Proving $(1+x)f'(x)=a f(x)$ where $f(x)=\sum_{n=0}^\infty {a\choose n}x^n$

A function $f$ is defined for $-1<x<1$ by $f(x)=\sum_{n=0}^\infty {a\choose n}x^n$. Here $a$ is a real number which is neither zero nor a positive integer. Prove that $(1+x)f'(x)=a f(x)$. I took ...
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1answer
25 views

Binomial coefficient (calculate m) [closed]

The problem asks: Calculate m knowing that $\left(\begin{matrix}m \\1 \end{matrix}\right) + \left(\begin{matrix}m \\2 \end{matrix}\right) + \left(\begin{matrix}m \\3 \end{matrix}\right)+ \ldots +\left(...
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1answer
45 views

Use combinatorial reasoning to show that Stirling number

Use combinatorial reasoning to show $\begin{Bmatrix} n\\ n-2 \end{Bmatrix} = \binom{n}{3} + 3\binom{n}{4}.$ The Stirling number is the number of permutation of n into $n-2$ parts.
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1answer
48 views

Calculating $\sum_{k=0}^{n}\binom{n}{k}\frac{1}{k+1}$

Hello everyone how can I calculate the sum of: $\sum_{k=0}^{n}\binom{n}{k}\frac{1}{k+1}$ I tried to use pascal identity and the binomial theorem and didn't success. Someone can help me please?
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0answers
33 views

Proof of $a^n<(a+h)^n<a^n+2nha^{n−1}$ for $h>0$ and sufficiently small $h$

I was reading this answer for the proof of $$\lim_{x\to a}{x^n}=a^n\ \forall \ n \in \mathbb{R}$$ Now..according to the answer given, the proof uses the following inequality $$a^n<(a+h)^n<a^n+...
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0answers
25 views

Using telescoping series to express a partial binomial series

I am analying $$ \sum_{j=k}^{n}\frac{(-1)^{j}{\frac{-3}{2}\choose n-j}}{j(j-1)}\;\;\mbox{where}\;n\ge k\;\mbox{are integers}. $$ By the telescoping series, we have $$ \sum_{j=k}^{n}(-1)^{j}{\frac{-3}{...
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1answer
32 views

How to prove the closed form for the generating function $g_r(x) = \sum_{n=0}^\infty {n \choose r} x^n$ possibly using Newton's Binomial Theorem

define the generating function $g_r(x) = \sum_{n=0}^\infty {n \choose r} x^n$ how do you find the closed form for this function? I got $x^r{(1-x)}^{-(r+1)}$ but I am not sure how to prove it. I ...
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2answers
27 views

Binomial coefficient after expansion

I am trying to solve an exercise where in the final step, I need to find the coefficient of $x^7y^5$ in $(x+y)^{12} + 7(x^2+y^2)^6 + 2(x^3+y^3)^4 + 2(x^4+y^4)^3 + 2(x^6+y^6)^2 + 4(x^{12}+y^{12}) + 6(x+...
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1answer
28 views

Binomial series: why do we take $n \cdot (n-1)$ out?

Given: $\sum_{k=1}^n \binom{n}{k} k(k-1)$. Find its closed form We can multiply $(k^2-k)$ with $\binom{n}{k}$ and get: $\frac{n!}{(k-2)!(n-k)!}$ Why do we take $n*(n-1)$ out: $n*(n-1)$*$\frac{(n-2)!...
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2answers
49 views

Prove combinatorics equality?

Assume $j$ is fixed, prove the following: $$\sum_{i}\binom{n}{i, j, n-i-j} = 2^{n-j}\binom{n}{j}$$ So the left hand side reminds me the multinomial theorem and we can think of a long sequence word ...
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3answers
50 views

Show that $n^5-n$ is divisible by 80 given that n is an odd integer greater than 1. [duplicate]

I tried to prove it by simple binomial expansion but I am stuck. I assumed, $n=(2k+1) ,k=1,2,3,......$ and after expanding the expression $(2k+1)^5-(2k+1)$ I arrived at this expression$→$ $8k+40k^2+...
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0answers
24 views

Proof of binomial theorem for non-integers

Firstly, yes, I know how to expand expressions like $(a+b)^{({1}/{2})}$ but I want a rigorous proof of why it is okay to replace binomial coefficients like ${1/2}\choose{2} $ with $({1/2})({1/2-1}))/2$...
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1answer
37 views

Prove $\binom{n}{k_1,…,k_m} = \sum_{i=1}^m \binom{n - 1}{k_1,…,k_{i - 1},..,k_m}$

I have a question ask to prove $$\binom{n}{k_1,...,k_m} = \sum_{i= 1}^m \binom{n - 1}{k_1,...,k_{i - 1},..,k_m}$$ I'm not sure how to approach this question, but the only thing that I noticed the LHS ...
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5answers
175 views

Factorise $x^4+y^4+(x+y)^4$

I think this problem involves making use of symmetry in some way but I don't know how. I expanded the $(x+y)^4$ term but it din't help in the factorization. I am very bad at factorizing, so I didn't ...
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4answers
42 views

Find the coefficient of $x^{-2}$ in the expansion of $(x-1)^3(\frac{1}{x} +2x)^6$

Find the coefficient of $x^{-2}$ in the expansion of $(x-1)^3(\frac{1}{x} +2x)^6$. I have tried finding out the general term of the whole expression. Then, I get the result as such $$[(-1)^r {3 \...
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1answer
26 views

$\sum_{r=0}^{n-k} {n \choose r}{n \choose {r+k}}={n \choose {n-k}}$ [duplicate]

$$\sum_{r=0}^{n-k} {n \choose r}{n \choose {r+k}}={n \choose {n-k}}$$ Can you prove why this result is as such? I have no idea how to start proving this property. Thank You!
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0answers
51 views

Determining the minimum score to pass

I am revising some basic statistics problems for my studies and came across this problem: A population of 10 000 students is taking an exam and the exam score is follows approximately a normal ...
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1answer
82 views

A Conceptual Intepretation of the identity

I came across the identity $$\sum_{k = 0}^{n}\frac{(-1)^{k}\binom{n}{k}}{2k+1} = \frac{2^{2n}(n!)^2}{(2n+1)!}$$ I tried it using the binomial theorem and integration and was able to prove it provided ...
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3answers
40 views

Sum of a binomial sequence equation

Anyone know how to go about solving this? I've tried adding the individual terms with no luck, and also I don't get how the 'n' in 15Cn doesn't correspond to (-1)^n ...
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1answer
54 views

Is there a formula for $(a+b)^n+(a-b)^n$?

Is there a formula for $(a+b)^n+(a-b)^n$? Just curious. Thanks.
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5answers
63 views

proof $k!j!\leq(j+k)!$ by induction [duplicate]

how do I prove that $k!j!\leq(j+k)!$ I have tried to use induction but didn't succeed. any other ideas on how to prove it? in witch case $k!j!=(j+k)!$ ?
5
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2answers
90 views

Proving interesting identity with partial sums of Pascal's triangle rows

As part of another problem I'm working on, I find myself needing to prove the following: $$\sum_{k=0}^n\binom{2k+1}{k}\binom{m-(2k+1)}{n-k} = \sum_{k=0}^{n}\binom{m+1}{k}$$ where $n\leq m$. I've ...
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1answer
30 views

Find coefficients of $x^2$ and $x^3$ in the expansion of $(3 − 2x) ^6$

The coefficients of $x^2$ and $x^3$ in the expansion of $(3 − 2x) ^6$ are $a$ and $b$ respectively. What is the value of $a$ and $b$? They need to be good at the binomial theorem and know the ...

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