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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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Summation of series to n terms in trigonometry of complex numbers

The question says that: Sum the series I have solved the answer as follows: As the above picture, I don’t know what should I do after the step. The question asks to solve the problem using ...
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1answer
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Is my proof of $\binom{m+n}{r}=\binom{m}{0}\binom{n}{r} + \binom{m}{1}\binom{n}{r-1} + … + \binom{m}{r}\binom{n}{0}$ right?

As the title says, I was requested to prove $\binom{m+n}{r}=\binom{m}{0}\binom{n}{r} + \binom{m}{1}\binom{n}{r-1} + ... +\binom{m}{r}\binom{n}{0}$ I was requested to do this using the following ...
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Help on proof of $\binom{n}{0}^2 + \binom{n}{1}^2 + … + \binom{n}{n}^2 = \binom{2n}{n}$

The proof is required to be made through the binomial theorem. I will expose the demonstration I was tought, and forward my questions after exposing it. You'll see question marks like this one (?-n) ...
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0answers
84 views

Summing Up Binomial Coefficients [duplicate]

$$ \frac{\sum_{r=0}^{24}\binom{100}{4r}\binom{100}{4r+2}}{\sum_{r=1}^{25}\binom{200}{8r-6}} $$ I tried this problem using Complex number approach and arrived at the solution. When I tried the ...
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1answer
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What is the intermediate step in this negative binomial proof?

On Slide 47 of these slides, there is a formula stating $$ s_k(t) = - \sum_{j=0}^{k} s_j(0) \sum_{n=j}^{k} \frac{e^{-\lambda_n t} \prod_{m=j}^{k-1} \lambda_m }{\prod^{k}_{m=j, m \neq n} (\lambda_m - \...
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0answers
30 views

Simple Proof of Binomial Theorem for Negative Integer Powers

There's a vast amount of clutter on the internet about this which I've been trawling through but it does not answer exactly what I'm asking which, because superficially similar questions have been on ...
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2answers
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To find the sum: $\frac {1}{n!} \sum \binom {n}{2+3r} x^{1+r}$

Sum the series: $$ \frac {x}{2!(n-2)!}+\frac {x^2}{5!(n-5)!}+\frac {x^3}{8!(n-8)!}+....+\frac {x^{\frac {n}{3}}}{(n-1)!}, $$ $n$ being a multiple of $3$.(Math. Tripos, 1899) My attempt We may ...
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How to code for $\phi_{k,N,L}=\sum_{i=k-L+1}^{k}\frac{\phi_{i,N-1,L}}{(k-i)!}I_{[0,(N-1)(L-1)]}(i)$

I have the following equation $$ \left[\sum_{k=0}^{m-1}\frac{1}{k!}x^k\right]^n=\sum_{k=0}^{n*(m-1)}\phi_{k,n,m}\times x^k. $$ where $\phi_{k,N,L}$ function define by $$\phi_{k,N,L}=\sum_{i=k-L+1}^{...
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How to simplify this sum

I'm having a lot of issues with this question. Ive done problems where I need to find the coefficient or the constant using the binomial theorem but I'm not sure how to even begin doing this. I looked ...
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1answer
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Using the binomial expansion to approximate a square root.

Find the first four terms of the expansion $\sqrt{1-4x}$ in ascending powers of x. Hence, approximate $\sqrt{6}$ to four decimal places. I've expanded it to form this: $1-2x-2x^2-4x^3$ When ...
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1answer
38 views

Binomial probability question!

How do I solve this question? A school lab has sixteen computers. A teacher observes that, in the long run, in 80% of school days, at least 1 machine is not working properly. Assuming the probability ...
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1answer
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Combinatorial identities, check the asymptotic behavior.

Check asymptotic of C = $\sum_{k = 0}^{\frac{n}{2} - \sqrt{n}} k \binom{n}{k} = f(n) + O(g(n))$ In the beginning I tried to simplify the expression under the sum: $$k\binom{n}{k} = k \frac{n!}{k!(n-...
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1answer
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Are there any power identities which don't belong to this list?

The problem of finding expansions of monomials, binomials etc. is classical and there is a lot of beautiful solutions have been found already, the most prominent examples are Binomial Theorem, ...
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If $C_0, C_1, C_2, .., C_n$ are the binomial coefficients in the expansion of $(1+x)^n$

If $C_0, C_1, C_2,...,C_n$ are the binomial coefficients in the expansion of $(1+x)^n$, prove that: $$C_{r}.C_{n} + C_{r+1}.C_{n-1} +......+ C_{n}.C_{r} = C(2n, n+r) =\dfrac {(2n)!}{(n-r)! (n+r)!}$$ ...
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1answer
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Need help expanding this negative binomial expanison

I am an A level Student studying for my papers and this is a question from a past paper...the original equation is $$\frac{1}{\sqrt{1+x}+\sqrt{1-x}}$$ which part a says to convert to the following ...
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1answer
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Evaluate: $\sum_{j=0}^{k}\sum_{i=0}^{j}{k \choose j}^2{k \choose i}$

Can anybody help me to evaluate this sum $(1)$? $$\sum_{j=0}^{k}\sum_{i=0}^{j}{k \choose j}^2{k \choose i}\tag1$$ I have manage to figure out: $$\sum_{j=0}^{k}\sum_{i=0}^{j}{k \choose j}{k \choose ...
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Number of ways to express sum.

Consider three sets of cards colored Blue, Red and Yellow. Each set has cards numbered $1-10$. The $4$ remaining cards are all indistinguishable cards numbered $0$. Card numbered $i$ has the ...
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4answers
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coefficient $x ^ n$ in development

The advisor asks to verify that the coefficient of $$x^n$$ in the development of: $$(1+x)^{2n}+x(1+x)^{2n−1}+x2(1+x)^{2n−2}+......+x^n(1+x)^n$$ is equal to $$\binom{2n+1}{n}$$ I tried for summations ...
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1answer
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Proof of the following combinatorial identity

Prove $$\sum_{m\le i\le k-l} \binom{k-i-1}{l-1} \binom{i-1}{m-1} = \binom{k-1}{m+l-1}$$ where $m$, $l$, $i$, $k$ are positive integers and $k\ge m+l$. Is this related to Vandermonde's identity?
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Proving $\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$

How can I prove this? $$\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$$ $$ 0\le k \le n $$ I developed the expressions, but they are not the same. I do not know if it will be ...
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1answer
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Why is this equality with sums and monomials true?

Consider $$(x_1 + ... + x_n)^k = \sum_{|\alpha| = k}c_{\alpha}x^{\alpha}$$ where $x^{\alpha} = x_{1}^{a_1}\cdots x_{n}^{a_n}$ and $|\alpha| = a_1 + ... + a_n$. Why is this true? Is it something to do ...
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7answers
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How to find divisibility of a very large number

I was asked this question in a test: The number $111111...111$ ($1$ comes 91 times) is a: A) Prime number B) Composite Number C) divisible by $\frac{10^7 -1}{9}$ The answer is B ...
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0answers
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strange binomial coefficient identity

Can someone please show why the following holds: $$ \sum_{k=0}^n\binom{n-1}{k-1}p^{k}q^{n-k} = p \sum_{k=1}^n\binom{n-1}{k-1}p^{k-1}q^{n-k} $$ Thanks in advance. Edit: Thank you for commenting. ...
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1answer
43 views

Combinatorial proof for $\sum_{i=k}^n (2i-k) \binom{i-1}{k-1}^2 = k \binom{n}{k}^2$

Give combinatorial proof for: $\sum_{i=k}^n (2i-k) \binom{i-1}{k-1}^2 = k \binom{n}{k}^2$ RHS: We want to make sequence A and B with their element only 0 and 1, and the number of element 1 is k in A ...
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1answer
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Find $a^3+b^3+c^3-3abc$ (binomial theorem) [duplicate]

$$a=\sum_{n=0}^\infty\frac{x^{3n}}{(3n)!}\\b=\sum_{n=1}^\infty\frac{x^{3n-2}}{(3n-2)!}\\c=\sum_{n=1}^\infty\frac{x^{3n-1}}{(3n-1)!}$$Find $a^3+b^3+c^3-3abc$: $(a)\ 1$ $(b)\ 0$ $(c)-1$ ...
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1answer
50 views

Combinatorial Proof (Argument)

Give a combinatorial proof for: $$\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^{k} \binom{n-1-j}{k-j} 2^j$$
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4answers
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Show that $\lim [(2n)^\frac{1}{n})] = 1$

Prove that $\lim ((2n)^\frac{1}{n}) = 1$. I have obtained the following: $$(2n)^\frac{1}{n} = 1 + k_{n}; n > 1$$ $$(2n) = (1 + k_{n})^n$$ By the Binomial Theorem: $$(1 + k_{n})^n = 1 + k_{n} + \...
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3answers
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Show that $\sum_{r=0}^{n} \binom{n}{r} r p^r q^{n-r} = np$, given $p+q=1$

How can I prove $$\sum_{r=0}^{n} \binom{n}{r} {r} p^r q^{n-r} = np,$$ given that $p+q=1$? I think it is about applying $$(1+x)^n=\sum_{r=0}^{n} \binom{n}{r} x^r.$$ Although I am quite unsure about ...
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1answer
48 views

Domain limitations on generating function for Legendre polynomials

The generating function for legendre polynomials is $$\frac{1}{\sqrt(1-2ut+u^2)}=\sum_{i=0}^\infty u^iP_i(t)$$ where $P_i$ is $i^{th}$ legendre polynomial however for binomial expansion of left hand ...
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2answers
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Binomial series expansion of a trinomial?

In electrostatics, the potential of a charge $q$ placed on the $z$-axis at $z=a$ is \begin{equation} \phi=\frac{1}{4\pi \epsilon_0}\frac{q}{r_1} \end{equation} where $r_1$ is the distance from the ...
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0answers
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$x^{2}+x+1$ is a factor of $(x+1)^{n} - x^{n} - 1$, whenever [duplicate]

Q) $x^{2}+x+1$ is a factor of $(x+1)^{n} - x^{n} - 1$, whenever (A) $n$ is odd (B) $n$ is odd and a multiple of $3$ (C) $n$ is an even multiple of $3$ (D) $n$ is odd and not a multiple of $3$ ...
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1answer
47 views

Closed form of the sum of two binomial expansions

I would like to know if there is a simple closed form for the following expression: $(x+y)^n + (x+z)^n$ Expanding the above I get $(y^n + z^n) + nx(y^{n-1}+z^{n-1}) + \frac{(n-1)n}{2}x^2(y^{n-2} + ...
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3answers
26 views

Binomial Expansion based on equation for evaluation

I have this question that has really stumped me, it is supposed to be done via Binomial kind of expansion. If $x+\frac1x=10$ find the value of $x^3+\frac1{x^3}$. So I hope some one has an approach ...
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2answers
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Newtons Binomial Theorem identity

I have seen a lot of identites being discussed here but I still haven't seen the one I'm having a problem with. I need to conclude that $\sum\limits_{i=1}^{n} i\binom{n}{i} \frac{(-4)^{i-1}}{5^{i-n}} ...
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3answers
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If $(1+x)^{n}=C_{0} + C_{1} x + C_{2} x^2+…C_{n} x^n$

If $(1+x)^{n}=C_{0} + C_{1} x + C_{2} x^2+......C_{n} x^n$, prove that: $$\textrm {a}. C_{0}C_{n}+C_{1}C_{n-1}+..........+C_{n}C_{0}=\dfrac {{2n}!}{{n}! \cdot {n}!}$$ $$\textrm {b}. {C_{0}}^{2}+{C_{1}...
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1answer
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$\int\frac{dx}{x(x+1)(x+2)\cdot\space…\space\cdot(x+n)}$ [duplicate]

I've been trying to solve explicitly the following indefinite integral: $$\int\frac{dx}{x(x+1)(x+2)\cdot\space...\space\cdot(x+n)}$$ I tried to perform partial fraction decomposition, and after ...
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2answers
41 views

Showing an Inequality Holds True

If $x$ is a positive real number and $n$ is a positive integer, prove the inequality, $\sqrt[n]{1+x}-1 \le \frac{x}{n}$. I tried to show this was true by doing the following: For $n = 1$, $\sqrt[1]{...
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1answer
42 views

How to evaluate a binomial sum with $2n$ in the exponent.

The question is to evaluate the sum $$\sum_{k=0}^n\binom nk 3^{2n-k}$$ have tried fitting into the binomial form of $\binom nk \times x^k\times y^{n-k}$ but I can't seem to bring it to the correct ...
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For which $k$ the following equation has the greatest value: $ k \cdot\binom{99}{k} $?

After some manipulation I got $$\frac{99!}{(k-1)!\cdot(99-k)!} $$ So I guess I have to find $k$ for which I get the smallest denominator, but I don't know where to go from there.
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1answer
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Maximum value of coefficient in Multinomial Expansion

Find the maximum value of coefficient in the expansion of $(x+y+z+w)^{25}$. Basically what the question is saying is that all term will be of type $k x^{r_1}y^{r_2}z^{r_3}w^{r_4}$ so what can be ...
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0answers
82 views

Finding a sequence with some summation properties $\sum\binom{n}{k}a_k=\binom{2n}{n}$

I was wondering if there exists a nontrivial sequence of numbers $a_{k}$ with the following property: $$\sum_{k=0}^{N}\binom{n}{k}a_{k}=\binom{2n}{n},$$ where the integer $N=\lfloor\tfrac12n\rfloor$ ...
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1answer
132 views

Prove that $16 ^ {2023} + 1$ is divisible by $17 ^ 2$.

Prove that $16 ^ {2023} + 1$ is divisible by $17 ^ 2$. It is clear that $16 ^ {2023} + 1$ is divisible by $17$, but why it is divisible by $17 ^ 2$ is not clear.
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6answers
177 views

Simple proof that $(1 + 1/n)^n$, $n \in \mathbb N$, is bounded above?

Here is the context of the problem. I'd like to prove that the sequence $$a_n = \left( 1 + \frac 1n \right)^n$$ converges using the monotone convergence theorem. It is straightforward using Bernoulli'...
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2answers
172 views

Why does the Binomial Theorem use combinations and not permutations for its coefficients?

I have been trying to understand the Binomial Theorem formula. I can see that it works. What I don’t understand is how or why using combinations finds the coefficients. What I mean is, isn’t each ...
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3answers
253 views

Question related to Pascal Triangles and Counting Patterns

I was working on two separate problems and came across a potential pattern involving Pascal's triangle. I would like to know if Pascal's triangle applies to either of these problems and if so, is ...
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3answers
101 views

In the expansion of $a{x}^3 {(2+ax)}^{11}$ , the coefficient of the term in $x^5$ is $11880$. Find the value of $a$ .

This is a question from the ib question bank which has the answer however the method isnt explained well enough. Its binomial theorem. Can someone explain the method in detail of how to deal with ...
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1answer
39 views

Proving the connection between the Binomial Theorem and the product rule for derivatives

Let $a(x)$ and $b(x)$ be smooth functions, i.e they are infinitely times differentiable. I have made the assumption that the derivative for the function $$f(x)= (a\cdot b)(x)$$ can be given by $$...
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3answers
91 views

How can I find the coefficient of $x^6$ in $(1+x+\frac{x^2}{2})^{10}$ efficiently with combinatorics?

To find the coefficient of $x^6$ in $(1+x+\frac{x^2}{2})^{10}$, I used factorization on $(1+x+\frac{x^2}{2})$ to obtain $\frac{((x+(1+i))(x+(1-i)))}{2}$, then simplified the question to finding the ...
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3answers
159 views

How can I prove $\frac {d}{dx} {x^n} = n x^{n-1}$ for $ n \in \Bbb R$ without circular reasoning? [duplicate]

I just cannot prove that $$\frac {d}{dx} {x^n} = n x^{n-1}$$ for $ n \in \Bbb R$. For $n \in \Bbb{N}$, I can use the definition of a derivative : $$\frac {d}{dx}x^n = \lim_{h \rightarrow 0} \frac{(...
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2answers
41 views

Simplifying an expression with binomial coefficients and powers of $2$

$$\binom{n}{0} \cdot 2^n + \binom{n}{1} \cdot 2^{n-1} + \binom{n}{2} \cdot 2^{n-2} + \dots + \binom{n}{n} \cdot 2^{n-n}$$ Anyway to simplify this such that it can become of 'closed' form (i.e. a ...