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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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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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Induction based on combinations and binomial theorem

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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1answer
39 views

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
53 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
36 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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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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Finding term that is independent of x in the expansion [closed]

Find the Expansion of $(2+3x^{-2})(x-2x^{-1})^6$ Can you expain it in detail? Struggling with this and exams are nearing it's urgent!
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1answer
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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{...
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3answers
60 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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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
49 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 ...
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1answer
43 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
40 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?
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2answers
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Proof of Inequality by Binomial Theorem

I have been stuck in this proof for a while. This should not be a long shot. Claim If $x_n>0$ for all $n\in N$, then $1+nx_n\leq (1+x_n)^n$. My strategy is to use Induction Theorem and Binomial ...
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2answers
64 views

How to solve interest problem without using a crazy binomial expansion?

Problem 9.9 Eric deposits X into a savings account at time 0, which pays interest at a nominal rate of i, compounded semiannually. Mike deposits 2X into a different savings account at time 0, ...
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2answers
59 views

In how many ways a student can get $2m $ Marks

An examination contains four Question papers each paper carrying maximum marks as $m$. Find number of ways a student appearing for all the four papers gets a total of $2m$ Marks. I used generating ...
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1answer
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Having trouble with a Binomial proof by mathematical induction question: $\sum_{j=3}^n \binom{j-1}{2} = \binom{n}{3}$ [closed]

I can't work out how to prove this equation is true by proof of mathematical Use mathematical induction to prove that, for $n \ge 3$ $$\sum_{j=3}^n \binom{j-1}{2} = \binom{n}{3}$$ Please help, ...
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Show $\left(1-\frac x k\right)^k<\left(1-\frac {x}{ k+1}\right)^{k+1}$

Show that $$\left(1-\frac{x}{k}\right)^k<\left(1-\frac{x}{k+1}\right)^{k+1}$$ for $x>0$, and $k \ge 1$, where $k$ is a whole number. Is it possible to prove this? I can easily prove ...
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2answers
56 views

Combinatorial proof of Negative Binomial Identity

For the (usual) Binomial theorem with positive integer exponent, there is a well known nice combinatorial proof. I am eager to learna similar argument for the proof of negative binomial series: $$(1+...
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2answers
34 views

Why binomial expansion approximation works?

So I have got the expansion of $$ (4-5x)^.5 = 2 + (5/4)x + (25/64)x^2 $$ I am told to use $ x = 1/10 $ to find an approximation of $ \sqrt2 $. I can do this, giving $ 181/128 $, however the last part ...
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343 views

Proof that $\sum\limits _{n=1}^{\infty } n\sum\limits _{j=2}^{\infty }{\frac {(-1)^{j-1}}{j^2} \left( 1-{j}^{-1} \right)^{n-1}} = -\frac12$

How can we prove the following? $-1+\frac1{12}{\pi }^{2}-\frac12\sum\limits_{n=2}^{\infty }\Gamma \left( n+1 \right) \sum\limits_{k=0}^{n+2} \,{\frac {\zeta \left( k \right) \left( - 1 \right)...
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References for $ \chi(n)=n\sum\limits_{j=2}^\infty\frac {(-1)^{j-1}}{j^2}\left(1-j^{-1}\right)^{n-1}$ in $\zeta$ expansion?

What is the name of these coefficients related to a series expansion for the Riemann zeta function or any other references? Did the author of this paper derive the series or get it from somewhere else?...
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1answer
57 views

How to simplify $\sum _{l=1} ^{k} \sum_{j_1 \ge 1} \cdots \sum_{j_l \ge 1, j_1 + \cdots + j_l=k} \binom{k}{j_1 \cdots j_l} \binom{-2}{l} 2^l$

I'm trying to simplify the following equation. $$\sum _{l=1} ^{k} \sum_{j_1 \ge 1} \cdots \sum_{j_l \ge 1, j_1 + \cdots + j_l=k} \binom{k}{j_1 \cdots j_l} \binom{-2}{l} 2^l$$ Or more specifically, I'...
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How to get the following expression ? 4 [closed]

Given the following expression: $$ f=\frac{1}{q (p+\sqrt{q})}\frac{1 + \exp(-2 \sqrt{q})}{1+ \frac{p - \sqrt{q}}{p + \sqrt{q}}\exp(-2\sqrt{q})} $$ I want to derive the following expression: $$ f = \...
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1answer
59 views

Coefficient of $x^k$ in ${(1 - e^x)}^{-n}$

What is the coefficient of $x^k$ in ${(1 - e^x)}^{-n}$? This is what I tried- Using negative binomial expansion and Taylor series expansion of $e^t$, $${(1 - e^x)}^{-n} = \sum_{i=0}^{\infty} {n+i-1 ...
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1answer
40 views

Simplify and compute the MGF of $[1-(1-(1-e^{-ax})^{N_1})(1-(1-e^{-ax})^{N_2})]^{M-1} $

Let $X_1$ and $X_2$ two random variable with CDF \begin{align} F_{X_1}(x)&=(1-e^{-ax})^{N_1}\\ F_{X_2}(x)&=(1-e^{-ax})^{N^2} \end{align} Let $Z$ random variable with CDF $$ F_Z(z)=[1-(1-F_{...
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2answers
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Derivation of Binomial theorem

Let $F(x)$ be the function defined by (1): $$ 1)F(x)=(1-e^{-ax})^N$$ Using the binomial theorem $F(x)$ can be written as (2): cc\begin{equation} 2)F(x)=\sum_{n=0}^{N}\binom{N}{n}(-1)^ne^{-axn} \end{...
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Question wrt Binomial theorem

This question may sound too stupid. But I am quite confused by the Binomial theorem As per my understanding, let us consider $(x+y)^4$ without the coefficients. $(x+y)^4 = x^4 + x^3y + x^2y^2 + xy^...
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Evaluating the variance of the biinomial distribution directly

I know that the easy way to evaluate the mean and variance of the Binomial distribution is by considering it as a sum of Bernoulli distributions. However, I was wondering just for fun if there is a ...
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5answers
62 views

Problem using binominal theorem? [closed]

I tried to solve this problem but I couldn't so i'm looking for an help. Is there any two digit natural number $n$ which fits with following statement? $$n \mid (4^n - 3^n - 1)$$ The hint is ...
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770 views

Remainder on division with $22$

What is the remainder obtained when $14^{16}$ is divided with $22$? Is there a general method for this, without using number theory? I wish to solve this question using binomial theorem only - maybe ...
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1answer
32 views

A curve C has an equation of $y = f(x)$, where $ f(x) = \frac{4x}{\sqrt{4+x}+\sqrt{4-x}} $, and $ -4\le x\le4. $

This question is Part $A$ of a binomial theorem question. I am unable to continue as I am stuck here. I really appreciate any help. A curve $C$ has an equation of $y = f(x)$, where $$ f(x) = \...
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1answer
59 views

Multinomial theorem & combinatorial probabilities

A set of $n$ people are given some sweets. There is candy A, B and C and each is given with probability $p_A,p_B$ and $p_C$. I am trying to find the possible combinations of this system. We can ...
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1answer
40 views

Is it possible to extract a variable out of a bounded sum of binomial coefficients

I've got a problem that looks like $$y = \sum\limits^x_{i=0} { a \choose i } b^i$$ where $a, b$ are constant. I'd like to rewrite this equation to define $x$ in terms of $y$, to directly compute the $...
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2answers
42 views

The relationship between the Pascal's triangle/sequence and the binomial number theorem?

What is the relationship between Pascal's sequence and the binomial theorem? I want to have a thorough and intuitive understanding of the connections between the two. Though I am able to relate to ...
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3answers
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What is the range of convergence of $\sum_{n=0}^{\infty} {(-1)}^n\binom{1/2}{n}\frac{1}{2n+3}.$

I was fiddling with the integral $$\int_0^1 x^2\sqrt{1-x^2} \ dx $$ and I expanded the term under square root using a binomial series. Integrating, I got the result $$\sum_{n=0}^{\infty} {(-1)}^n\...
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1answer
54 views

Prove the following Combinatorial Identity:

I actually would like to try and figure out a proof for myself, but I would like to ask if anyone could provide me a hint to successfully proving the following identity: $$\sum_{k = 0}^n k \binom{n+1}...
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1answer
20 views

Sum of independent Poisson distributions and binomial theorem

I have two independent Poisson variables $X_1$ and $X_2$ for which I have calculated the sum in convolution. I obtained: $$ e ^{\mu_1 \mu_2}\sum_{k=O}\frac{\mu_1^k}{k!}\frac{\mu_z^{z-k_1}}{z-k_1!}$$ ...
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3answers
43 views

Hint on how to prove the identity $\sum_{0 \leq k \leq n } \binom{k}{m} = \binom{n + 1}{m + 1}$

Let $N$ be a positive integer. Then, we have $\sum\limits_{j=1}^{N} \binom{j}{6} = \binom{N+1}{7}$. Could anyone explain this equation a little bit? I wrote out the left hand side as $\binom{1}{6} + ...
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31 views

Expansion question involving Taylor or Binomial series?

enter image description here I have tried to solve this question using binomial series, Taylor series but I seem to be heading nowhere… I seem to get lost every time I attempt it.
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Combinatorics Binomial How to prove it? [closed]

$$ \sum_{k=n}^{2n} \binom{k}{n} 2^{-k}=1 $$ Anyone can help me? How to prove it?
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61 views

Prove the Binomial Theorem using induction (What is mathematics book)

I read What is Mathematics book by Richard Courant and Herbert Robbins and there is an exercise where I should prove the Binomial theorem using induction. Here is an expression to be proved: $$ C^n_i ...
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1answer
45 views

How can I find a short term for those sums?

I have $A=[a_1,a_2,...,a_{2016}]$. The number of subsets of $A$ where the number of elements is divisible by $4$, is $M$. The number of subsets of $A$ where the number of elements is divisible by $2$ ...
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Proving $\sum\limits_{k=0}^{\infty}\binom {m-r+s}k\binom {n+r-s}{n-k}\binom {r+k}{m+n}=\binom rm\binom sn$

Question: How do you show the following equality holds using binomials$$\sum\limits_{k=0}^{\infty}\binom {m-r+s}k\binom {r+k}{m+n}\binom {n+r-s}{n-k}=\binom rm\binom sn$$ I would like to prove the ...
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1answer
117 views

How can I prove the following equality?

I need to prove: $$\sum_{k=0}^{n} {k\choose 2}\cdot{{n-k}\choose 2} = \sum_{k=0}^{n} {k\choose 3}\cdot(n-k).$$ I wasn't able to have any progress with algebra, I tried to think about a combinatorical ...
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3answers
42 views

Which one is greater (A or B)?

Q. If $$A=(99)^{50} + (100)^{50}$$ and $$ B=(101)^{50}$$ then- $(a) A>B$ $(b) A<B$ $(c) A=B$ My attempt - I thought of using the binomial approximation to A and B , which gives - $$ A= (100)^...
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5answers
135 views

How many digits is $99^{99}$ without a calculator?

I know the answer is $198$. I realise that if $\log _{10}\left( x\right) =y$, the number $x$ has $\lfloor y\rfloor -1$ digits So I tried $\log ^{\ }_{10}\left( 99^{99}\right) $ = $\log _{10}\left( ...
0
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1answer
31 views

Upper bound of of powers of a binomial

I need to proof a property of an object of statistic. I would have finished successfully if it was not for me to use an upper bound of a power of a binomial that i found in my book that I can't proof ...
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0answers
24 views

Proving the Hockey Stick Identity with the Summation Identity [duplicate]

The Summation Identity is: $\dbinom{n}{r}+\dbinom{n}{r+1}=\dbinom{n+1}{r+1}$. Is there a way to prove the hockey stick identity with this? Hockey Stick Identity: $\dbinom{n}{r}+\dbinom{n+1}{r}+\...
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1answer
40 views

How is $ C_0^n+\frac{1}{2}(C_0^n-C_1^n)+\frac{1}{3}(C_0^n-C_1^n+C_2^n)+.. =C_0^{n-1}+\frac{1}{2}C_1^{n-1}(-1)+\frac{1}{3}C_2^{n-1}(-1)^2… $? [closed]

I saw this result in the solution of a question: $ C_0^n + \frac{1}{2}(C_0^n-C_1^n) + \frac{1}{3}(C_0^n-C_1^n+C_2^n)+... = C_0^{n-1} + \frac{1}{2}C_1^{n-1}(-1) + \frac{1}{3}C_2^{n-1}(-1)^2... = 1/n $. ...