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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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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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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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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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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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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 ...
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Determine the co-efficient of $x^{97}$ in the expansion of $(1-x)^{135}(1+x+x^2)^{135}$.

Determine the co-efficient of $x^{97}$ in the expansion of $(1-x)^{135}(1+x+x^2)^{135}$. I know to use the binomial theorem, but I am having a difficult time simplifying. Any help is appreciated, ...
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Binomial Theorem expansion and proving an interesting identity?

In the identity $$\frac {n!}{x(x+1)(x+2)...(x+n)} = \sum ^n_{k=0}\frac {A_k}{x+k} $$ Prove that $$A_k =(-1)^{k}\:^{n}C_k$$ Also from this deduce that, $$ \;^{n}C_0\frac 1{1.2} - \:^{n}C_1\frac1{2....
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By considering expansions of (1+(1+x))^n and (2+x)^n, show that

(nCr)(rCr) + (nCr+1)(r+1Cr) + (nCr+2)(r+2Cr) + ... + (nCn)(nCr) = (nCr)2^(n-r) I have expanded both initial binomial expansions, but am unsure where to continue from here.
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equating coefficients in algebraic expansion

If $\displaystyle \bigg(\frac{1+x}{1-x}\bigg)^n=1+b_{1}x+b_{2}x^2+b_{3}x^3+\cdots +\infty,$ then value of $(1)\; \displaystyle \frac{3b_{3}-b_{1}}{b_{2}}$ $(2)\; \displaystyle \frac{2b_{4}-b_{2}}{...
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If $(A+B)^n$ is binomial for some $n$, does that imply $AB = BA$?

We say that a matrix power $(A+B)^n$ is binomial iff it satisfies the matrix equality $$(A+B)^n = \sum \limits_{j\,=\,0}^n \binom{n}{j}A^jB^{n-j}.$$ If two matrices have a binomial power for some $n$, ...
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On the parity of the coefficients of $(x+y)^n$.

The coefficients of $(x+y)^3=x^3+3x^2y+3xy^2+y^3$ are $1$, $3$, $3$ and $1$. They are all odd numbers. Which of the following options has coefficients that are also all odd numbers? $(\text A) \ \...
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Concerning the identity in sums of Binomial coefficients

Let be the following identity $$\sum_{k=1}^{n}\binom{k}{2}=\sum_{k=0}^{n-1}\binom{k+1}{2}=\sum_{k=1}^{n}k(n-k)=\sum_{k=0}^{n-1}k(n-k)=\frac16(n+1)(n-1)n$$ As we can see the partial sums of binomial ...
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Limit of a sum with binomial term, where terms do not add up to 1

I am trying to compute the limit of the following sum involving a binomial term, where $\mu\in[0,1]$ and $\theta\in[0,1]$: $ \displaystyle \lim_{t \to \infty} (1-\mu)(1-\theta) \displaystyle \sum_{r=...
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Summation of combination of binomial coefficient

Is there any way to find: $$\sum_{i=0}^n {\binom{n}{i}i^k}$$ I know that we can find it for small k by using binomial theorem by differentiating both sides and then multiplying both sides by x and ...
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Finding Remainder Using Binomial Theorem

Find the remainder when $7^{98} $ is divided by $5$. What I am doing here is expanding ${(5+2)}^{98} $ using binomial theorem and writing it as $5k + 2$, where $k$ is a positive integer but the ...
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series $\sum_{n=1}^\infty \sum_{m=1}^\infty \frac{(m+n-1)!}{m!(n-1)!n!(m-1)!}a^m b^n.$

Can anyone please help me with the computation of following series: $$\sum_{n=1}^\infty \sum_{m=1}^\infty \frac{(m+n-1)!}{m!(n-1)!n!(m-1)!}a^m b^n.$$ My thoughts: Since $$\displaystyle \frac{(m+n-1)!...
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Find the coefficient of $x$ in the given polynomial

We have polynomial $$(x-2^0)(x-2^1)(x-2^2)···(x-2^n)$$ I know that the coefficient of $x$ equals to: $$(-1)^{n-1}((2^0×2^1×···×2^{n-1})+(2^0×2^1×···×2^{n})+···+(2^1×2^2×···×2^{n}))$$ But it's hard to ...
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Closed form using Binomial Expansion

I would like to obtain a closed form for the following: $${n \choose 0}{n \choose k} + {n \choose 1}{n-1 \choose k-1} + {n \choose 2}{n-2 \choose k-2} + ... + {n \choose k}{n-k \choose 0}$$ To me, ...
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Summation using binomial theorem

Find the sum $$\sum_{x=1}^{m} \Big(\frac{2^x-1}{2^x}\Big)^n$$ where $n$ belongs to natural number.
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a combinatoric limit proof

Let $\beta >4$. Prove that if $$a_n = \sum_{k=1}^{n}k{n\choose k}{n\choose {k-1}}$$ then $$\lim_{n=0}\frac{a_n}{\beta^n} = 0$$ My attempt to solve it: I tried to show that $a_n \leq 4^n = \sum_{k =...
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A query in binomial th. Based question

Posting the part of the solution...i know the concept on which it is asked but i got stuck while solving and i peeked into the solution, i found this $=\frac{2n(2n-1)(2n-2)...4.3.2.1}{n!n!}x^n$ ...
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39 views

The number of distinct terms in a trinomial expansion

$$(x^3 + \frac1{x^3} +1)^{200}$$ is the given expression. How many distinct terms are in this expression when expanded. I know that there are a total of $3^{200}$ terms before combining the terms but ...
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62 views

Maximum coefficient in the expansion of $(5+3x)^{10}$

Though I know that I could simply just expand $(5+3x)^{10}$ with the binomial theorem for each power of x, is there a simpler and quicker method of finding out the largest coefficient? After manual ...
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A question on binomial theorem

If $C_0$, $C_1$, $C_2$,...$C_n$ are the coefficients in the expansion of $(1+x)^n$, where $n$ is a positive integer, show that $$C_1- {C_2\over 2} +{C_3\over 3}-...+{(-1)^{n-1} C_n\over n}=1+ {1\over ...
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How can I calculate how many times a given subset of (finite) X will appear in the set of subsets of X of a particular size?

Let $X$ be a finite set. Suppose $R$ is some proper subset of X. Let $S$ be the set of subsets of $X$ of a given size; for example, $S$ could be the set of three-membered subsets of $X$. Is there any ...
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Calculating number of times a given element appears in powerset subsets of a particular size.

Let $X$ be a finite set, and $P(X)$ the power set of $X$. I understand from here how to calculate the number of subsets of a particular size in $P(X)$. For a given member of $X$ (call it $n$) how can ...
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Using the binomial expansion to show $\sum_{k=0}^n {n \choose k} \frac{1}{2^k} = (3/2)^n$

I have to show $$\sum_{k=0}^n {n \choose k} \frac{1}{2^k} = (3/2)^n$$ using the binomial theorem. I haven't had any practice on these types of questions so i'm unsure as to how to proceed. Would ...
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How to find all $x \in \mathbb{Q}$ and $r \in \mathbb{Q}$ such that $(1+x)^r$ becomes a rational number?

Let $x, \ r \in \mathbb{Q}$. I need to find the conditions on $ \ x, \ r$ so that the value of $ \large (1+x)^r$ is a rational number. Which $x, \ r$ makes $(1+x)^r$ a rational number? Answer: ...
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1answer
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A.M of smallest elements of $r$ subsets of set 1,2,3,…,n

All possible subsets containing $r$ elements from the set$\{1,2,3,...,n\}$ are formed where $(1\leq r\leq n)$. What is the arithmetic mean of the smallest elements of these subsets. My Attempt The ...
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274 views

Trying to parse a Putnam solution from 1995

I’m having trouble parsing the solution for the 1995 Putnam, question A2. The proof proceeds: The easiest proof uses ``big-O'' notation and the fact that $(1+x)^{1/2} = 1 + x/2 + O(x^{2})$ for $|x|&...
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Evaluate a binomial-like sum

Let $x \in \mathbb{R}$ and let $n \in \mathbb{N}$ then evaluate: $$\sum_{k=0}^n{n \choose k}\sin\left(x+\frac{k \pi }{2}\right)$$ I could only go up to breaking this sum in two parts of $\sin x$ and $...
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Evaluate:$S_{n}=\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{n-3}-\binom{n-3}{n-6}+…$

If$$S_{n}=\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{n-3}-\binom{n-3}{n-6}+.......$$ Does $S_{n}$ have a closed form. My Attempt $$S_{n}=\binom{n}{0}-\binom{n-1}{n-2}+\binom{n-2}{2}-\binom{n-3}{3}+........
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Prove that $n^{n-1} - 1$ is divisible by $(n-1)^2$ [duplicate]

I am trying to solve the following question: For each positive integer $n$, strictly larger than 1, it holds that $n^{n-1}-1$ is divisible by $(n-1)^2$. This question appears in a chapter on the ...
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Why can't x be 1 in the binomial expansion for any index?

For the series to converge, $R_n$ must be zero when n tends to infinity. In equation (5.101), when m is a nonnegative integer and x=1, the only effective term left is $\frac{1}{n!}$ that becomes zero ...
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1answer
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Finding Laurent Series using Binomial Theorem - HOW?

I'm working on a fairly simple question asking to work out the necessary branch cut(s) for the function $f(z)=(z^2+1)^{1/2}$. I am comfortable doing this and the rigour required to explained why I ...
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Prove that $\sum_{k = 0}^{49}(-1)^k\binom{99}{2k} = -2^{49}$

I am preparing a class on the binomial of Newton. One of the exercises at the end of the chapter turns out to be very hard for me: Prove that $$\sum_{k = 0}^{49}(-1)^k\binom{99}{2k} = -2^{49}$$ I ...
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Combinatorial Proof for the equation $\sum_{i=0}^j {j \choose i} 2^{j-i} = 3^j$

$$\sum_{i=0}^j {j \choose i}2^{j-i} = 3^j$$ My approach: I know the binomial way to do this is to think of the RHS as $(1+2)^j$ and then expand using binomial like so: $$(1+2)^j = \sum_{i=0}^j {j \...
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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 ...
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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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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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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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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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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
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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
21 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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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
32 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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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
57 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 ...