Questions tagged [binomial-theorem]
For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.
0
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0answers
16 views
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 ...
3
votes
2answers
48 views
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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votes
1answer
33 views
How would I find an element or elements that has/have $x^5$ in trinomal development $(x+{1/x^{3}}+x^2)^8$ [on hold]
This is what i did.
$$\left(x+{\frac{1}{x^{3}}}+x^2\right)^8
=\left(\frac{(x^{4}+1+x^{5})}{x^3}\right)^8=\frac{1}{x^{24}}(x^4+1+x^5)^8$$
1
vote
2answers
37 views
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, ...
0
votes
1answer
18 views
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?
2
votes
2answers
21 views
Question on word combinations with exclusivity
"How many 4 letter words on the alphabet {a,b,c} in which 'a' occurs exactly twice are there?"
My answer is incorrect as I answered 3*3*2*2 4 letter words.
However, this doesn't necessarily remove '...
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2answers
28 views
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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votes
0answers
16 views
analytic integration for binomial term?
I have questions about method of analytic integration
I have an expression
$$p(N|a,s,b) = \frac{\exp[(-as+b)](as+b)^N}{N!} * \frac{\exp(-gs)(gs)^A}{\Gamma(A+1)}*\frac{\exp(-hb)(hb)^B}{\Gamma(AB+1)}$...
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vote
4answers
68 views
What is the coefficient of $x^{11}$ in $(3x-9)^{19}$?
I am currently studying for finals, and I do not know how to do this problem from my study guide. I have tried to watch a few YouTube videos and I know that I will end up with $3x^{11} \times (-9)^8$, ...
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1answer
16 views
Let n belongs to +ve integer and $(1+x+x^2)^n=\sum_{r=0}^{2n} {a_rx^r}$ prove that: $a_r=a_{0<r<2n}$
Let n belongs to +ve integer and $$(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)$$. I tried to ...
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votes
1answer
31 views
Complex number identity question
If $n$ is even and $z=\cos\theta+i\sin\theta$. By expressing $z^n$ in two ways, show that
$$\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
30 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
38 views
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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1answer
47 views
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 ...
0
votes
2answers
54 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 ...
1
vote
2answers
75 views
Weighted sum of product of binomial coefficients
I am trying to evaluate the sum
$\displaystyle \sum_{n=1}^N \sum_{k=1}^n k\binom{n}{k} \binom{N-n}{k}x^k$,
Here $x$ is some positive real
My approach so far has been to first to compute the ...
0
votes
0answers
253 views
Probability question: calculating probability of attendance in training camp.
Assume there are $400$ athletes in a training camp, who are required to attend the morning drill starting at $4$ am. The attendance in morning drills is $70\%$, i.e. on an average, $280$ athletes are ...
2
votes
5answers
63 views
How to show that $2^n > n$ without induction
I'm solving exercises about Pascal's triangle and Binomial theorem, and this problem showed up, however I don't have any clue on how to solve it
The sum of ${n\choose p}$ from $p=0$ to $n$ is the ...
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votes
1answer
53 views
Proving $\int ^{2\pi} _{0} (\cos(t)^{2n})={2n \choose {n}}\frac{2\pi}{2^{2n}}$
using the following result
$\int _{\gamma}(z+ \frac{1}{z})^{2n}\frac{dz}{z}= {2n \choose {n}}2\pi i $
Prove
$\int ^{2\pi} _{0} (\cos(t)^{2n})={2n \choose {n}}\frac{2\pi}{2^{2n}}$
I cant see how ...
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votes
4answers
65 views
I tried this question by using pi denoting method and got a big equation what to do
Find coefficient of $x^6$ in
$(1+x)(1+x^2)^2.....(1+x^n)^n$
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votes
1answer
28 views
How does it follow from the Pascal's Triangle that binomial coefficient are integers
So I was reading this lemma which states:
Let $m,n$ be natural numbers such that $1 \leq m \leq n$. Then
\begin{equation*}
{n\choose m-1} + {n\choose m} = {n+1\choose m}.
\end{equation*}
It follows ...
0
votes
0answers
16 views
Analytic integration of profile likelihood vs marginal likelihood?
I have questions about 1. Method of analytic integration 2. Statistics (marginal vs profile)
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)}*\...
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votes
0answers
32 views
Use the Binomial Formula to show that
There's the problem
$$
\binom{n}{0}+\binom{n}{2}+\cdots=\binom{n}{1}+\binom{n}{3}+\cdots=2^{n-1}
$$
I know how to do operations with the binomial formula but I do not understand what that set up is ...
2
votes
2answers
125 views
Strict Bernstein Inequality
We define the entropy function by
$$H(x)=x\ln\frac{1}{x}+(1-x)\ln\frac{1}{1-x} \quad \text{ for } 0 \leq x \leq 1 $$
where $H(0)=H(1)=0$.
a) For integers $0\leq k\leq n$ prove that
$$\...
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votes
0answers
91 views
Strengthening of Bernstein Inequality
Problem. Let $X_1, . . . , X_n$ be independent random variables such that $\mathbf{E}(X_k) = 0$ and
$|X_k| \leq 1$ for $k = 1, . . . , n$. Let $X = X_1 + . . . + X_n$. Prove the following ...
1
vote
1answer
31 views
Binomial Expansion rth term
Given that there's no term on $x^3$ in $$(k + 2x) (1 - 3/2 x) ^6.$$ Evaluate the value of $k$
Difficult
After expanding $(1 - 3/2 x)^6$ up to the 3rd term I get $1 - 9x + \frac{135}{2} x^2 - \frac{...
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votes
1answer
57 views
Showing that $\left(1+\frac{\varepsilon^2}{17n}\right)^n-1\leq \frac{\varepsilon^2}{16}$ when $0\leq \varepsilon\leq 1$
Question Show that $\left(1+\frac{\varepsilon^2}{17n}\right)^n-1\leq \frac{\varepsilon^2}{16}$ when $0\leq \varepsilon\leq 1$.
This inequality appeared in the middle of an argument I was reading and ...
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0answers
70 views
Proof of Newton's Generalized Binomial Theorem (without Calculus)
I'm writing an article for derivates, I've already prooved Newton's Binomial Theorem, but I want to proof that the expresion
$$(a+b)^r=\sum_{i=0}^\infty\binom{r}{i}a^ib^{r-i}$$
works for all $a,b,r\in\...
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votes
1answer
39 views
How to prove Binomial Theorem
I have got through part a) and b)
The answer is 365 and 364, respectively , however I'm not able to tackle part c)
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vote
1answer
39 views
Evaluate $lim_{n \to \infty} (n + 1)^{\frac{2}{3}} - (n)^{\frac{2}{3}}$ using the generalized binomial theorem
I have already checked this question and have understood the approaches using the Product Rule for Limits, and the one using the Mean Value Theorem.
I was wondering whether it is possible to evaluate ...
2
votes
2answers
43 views
How can I find the number of dissimilar or distinct terms in $(1+x + x^2 +x^3)^n$?
How can I find the number of dissimilar or distinct terms in $(1+x + x^2 +x^3)^n$ ?
I know it would be $\binom{n+r-1}{r-1}$ when they are $a , b, c ,d$ instead of $1 , x , x^2 , x^3$.
But how ...
1
vote
2answers
35 views
How to find the sum of the given series: $\sum_{k=0}^{\min(n ,m)} \binom{n}{k} \binom{m}{k}$?
I wonder whether it is possible to calculate the following sum that involves the Binomial coefficients
$\sum_{k=0}^{\min(n ,m)} \binom{n}{k} \binom{m}{k}$ .
4
votes
2answers
88 views
If $y=\frac25+\frac{1\cdot3}{2!}\left(\frac25\right)^2 +\frac{1\cdot3\cdot5}{3!}\left(\frac25\right)^3+\ldots$ then what is the value of $y^2+2y$?
If
$$y=\frac25+\frac{1\cdot3}{2!}\left(\frac25\right)^2 +\frac{1\cdot3\cdot5}{3!}\left(\frac25\right)^3+\ldots$$
then what is the value of $y^2+2y$?
This is a question from my coaching material in ...
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votes
2answers
59 views
How is $C_0^nC_n^{n+1}+ C_1^nC_{n-1}^{n}+C_2^nC_{n-2}^{n-1}+\dots+C_n^n C_0^{1} =2^{n-1}(n+2) ?$ [closed]
How is $$C_0^nC_n^{n+1}+ C_1^nC_{n-1}^{n}+C_2^nC_{n-2}^{n-1}+\dots+C_n^n C_0^{1} =2^n(n+2) ?$$
I have no idea how to approach this problem. There is no solution given in my book. There doesn't seem ...
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votes
1answer
29 views
Small induction problem.
Show with the help of induction for all natural numbers n, where a and b are real numbers and:
$a \neq b$ and $ a + b > 0$
$$(a + b)^n ≤ 2^{n−1}\cdot(a^n + b^n) $$
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votes
1answer
30 views
Trying to evaluate the constant term [closed]
Suppose you are given that
$$\biggr (\sqrt[3]x - \dfrac{1}{\sqrt x} \biggr )^{15}$$
I'm trying to evaluate the constant term.
$$\sum^{15}_{n = 0} \binom{6}{r}x^{5-r}\cdot -\dfrac{\sqrt x}{x}$$
...
2
votes
2answers
69 views
Calculating coefficient
I have a generating function,
$$ \frac{(1-x^7)^6}{(1-x)^6} $$
and I want to calculate the coefficient of $x^{26}$
Solution for this is,
$$ {26+5 \choose 5} - 6{19+5 \choose 5} + 15{12+5 \choose 5} - ...
2
votes
2answers
84 views
Numerical differentiation with Binomial Theorem
In George Shilov's Elementary Real and Complex Analysis, there is a problem which asks us prove
If $f$ is twice differentiable on some open interval and the second derivative is continuous at $x$, ...
0
votes
3answers
51 views
Find the coefficient of $x^3$ in the expansion of $(1+x+x^2)^3$ using Binomial theorem. [closed]
Find the coefficient of $x^3$ in the expansion of $(1+x+x^2)^3$
using Binomial theorem.
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votes
4answers
39 views
Intuitive understanding of the binomial theorem?
I'm trying to understand the thought process, of how I might come upon the binomial theorem intuitively by thinking about combinatorics, can someone help?
2
votes
3answers
45 views
Coefficient in binomial expansion [closed]
Whats is the coefficient in front of $x^{50}$ in the binomial expansion $$(1+x)^{1000} + (1 + x)^{999}\cdot x^1 + ... + (1+ x)^1\cdot x^{999} + x^{1000}$$
0
votes
1answer
25 views
2n-th derivation of Legendre Polynomial
Let $u_n(x)=(x^2-1)^n$
Show that $\frac{(d^{2n}u_n(x)} {dx^{2n}} = 2n!$
$(x^2-1)^n = (x-1)^n(x+1)^n $ and then a should use Leibnitz formula. I thought if I write Leibnitz formula as a binomial I ...
1
vote
2answers
90 views
Binomial sum of $\sum\limits_{k=0}^{n} k \binom{n}{k} (-1)^{k-1} 2^{n-k}$
Evaluate $\displaystyle{\sum_{k=0}^{n} k \binom{n}{k} (-1)^{k-1} 2^{n-k}}$
So if the $k$ wasn't here this binomial sum would be really easy to solve. How would you evaluate this then?
1
vote
3answers
30 views
What is the coefficient of the term $x^3y^5$, as a result of the binomial expansion of the following term?
We have the term
$(1+xy+y^2)^n$
If we expand it using the binomial theorem, why is the factor of the term $x^3y^5$ the following: $4{n\choose 4}$? (The binomial coefficient n choose 4 multiplied by 4)...
3
votes
2answers
42 views
Binomial Expansion to the power of a non-natural number
What is the reasoning that $|x|$ has to be less than $1$ for $(1+x)^n$ when $n$ is not a natural number?
4
votes
1answer
63 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 ...
1
vote
2answers
50 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 ...
0
votes
3answers
31 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)...
-1
votes
2answers
41 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 ...
0
votes
3answers
85 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 $\left(3-\frac1{9x}\right)^5(2+9x)^2$....
