Questions tagged [multinomial-theorem]
An extension to the binomial theorem. It gives the expansion of a multinomial $(x_0,\dots,x_{m-1})^n$.
0
votes
3answers
38 views
Coefficient problem in algebra
Find the coefficient of $ x^{8} $ in the expansion of $ (1+x^2-x^3)^{9} $
I know the problem is simple if we use multinomial theorem and I got an answer $ 378 $ using it. Can someone check it and ...
2
votes
1answer
26 views
The coefficient of $x^n$ in the expansion of $\frac{2-3x}{1-3x+2x^2}$ is?
The coefficient of $x^n$ in the expansion of $\frac{2-3x}{1-3x+2x^2}$ is?
Working: $1 – 3x + 2x^2 = (1 – x)(1 – 2x)
=> (1 – x)^{-1} = 1 + (-1)(-x) + {(-1)(-1 -1)/2!}(-x)^2 + {(-1)(-1-1)(-12)/3!}(-...
0
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0answers
15 views
How to analytically calculate a specific pseudoinverse for multinomial root finding
Consider the following problem:
$$d_1=a_{1,1}x+a_{1,2}y+a_{1,3}x^2+a_{1,4}2xy+a_{1,5}y^2$$
$$d_2=a_{2,1}x+a_{2,2}y+a_{2,3}x^2+a_{2,4}2xy+a_{2,5}y^2$$
and suppose we can represent it as follows:
$$\...
0
votes
4answers
47 views
if in the expansion of $(1+x)^m (1-x)^n$, the coefficients of $x$ and $x^2$ are $3$ and $-6$. Find value of $m$ and $n$. [closed]
If in the expansion of $(1+x)^m (1-x)^n$, the coefficients of $x$ and $x^2$ are $3$ and $-6$. Find value of $m$ and $n$.
Please help me solve the above problem
0
votes
1answer
35 views
Multinational Coefficient Difficulty
Given
$$\left( 2x + xy - z + \frac{1}{xyz} \right) ^{10}$$
it is asked to caltulate the coeficient of
$$ x^6 y^5 z$$
I tried to simplify the formula to only have 3 "parcels" but with no success. the ...
0
votes
1answer
21 views
Sum of dependent Binomial distributions
In one of my classes, we stated that if $X_i$ are independent Bernoulli random variables with p proportion of success, then the distribution of the sum $\sum X_i$ is Binomial(n,p). I already proved ...
1
vote
1answer
48 views
Simplifying summation of binomials
I am working on a proof and think that the following equality holds but am unable to prove it:
$$
\sum_{k_1=0}^{m} \sum_{k_2=0}^{m-k_1} \dots \sum_{k_{d^2/2}=m-(k_1+k_2+\dots+k_{d^2/2-1})}^{m-(k_1+k_2+...
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votes
0answers
23 views
Is it possile to apply the Multinomial Theorem when the sum inside the power is infinite and contains a general function of the sum parameter?
I hope the title communicates properly what I want to say and solve. I have the following expression
$$
\sum_{m=1}^{\infty}\frac{(-1)^{m-1}}{m}\left(\sum_{k=1}^{\infty}z^{k}A(k)\right)^{m}
$$
This ...
0
votes
0answers
27 views
Evaluating a sum involving multinomial coefficient
$$S(p,x,y)=\sum_{n=1}^{\infty}(1-p)^{n-1}\sum_{a+b\le n,0\le b<a}\binom{n}{a,b,n-a-b}x^ay^b(1-x-y)^{n-a-b}$$
I am unable to simplify the sum. Mathematica doesn't help either. Any help will be much ...
1
vote
1answer
31 views
How to prove $\sum_{\alpha_1=0}^k \sum_{|\alpha|=k} \dfrac{k!}{\alpha!} a^\alpha = \sum_{|\alpha|=k} \dfrac{k!}{\alpha!} a^\alpha$
In order to prove the multinomial theorem, I have to prove an intermediate result and have been stuck at the last step.
For $k,m \in \Bbb N$ with $m \ge 2$, $\alpha = (\alpha_1,\cdots,\alpha_{m+1}) ...
1
vote
3answers
49 views
Trinomial expansion with power constraints
For the trinomial expansion $(a+b+c)^n$, I'd like to sum up the terms like $a^i b^j c^k$ with the constraint $i>j$. How to calculate it efficiently?
2
votes
2answers
54 views
Finding sum multinomial
I did put x=$w, w^2 ,i ,-i$ but nothing of type is fetting
formed. How come 1/2 is remaining constant.
That means because of some substitution,
$2a_o= a_1+ a_2$ is happening.
Also tried putting x=ix.
0
votes
1answer
60 views
Why can’t we use multinomial theorem here?
We have $10$ white, $9$ green and $7$ black balls. All balls are identical except for colour. While the solution for selecting number of ways in which one or more balls can be selected from these ...
-2
votes
2answers
45 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.
1
vote
2answers
100 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
66 views
Sum of coefficients in multinomial expansion
If $x,y,z$ are independent of each other, then the sum of the coefficients in the expansion of $(5x+3y-8z)^{30}$ is -
0
votes
1answer
77 views
Expansion of the cube of the sum of N numbers
I know that the expansion of the square of a summation can be expressed as:
$$\left( \sum_{n=1}^N a_n \right)^2 = \sum_{n=1}^N a_n^2 + 2 \sum_{j=1}^{N}\sum_{i=1}^{j-1} a_i a_j$$
where $a_n \in \...
0
votes
1answer
28 views
Using Bayes' with a Multinomial Distribution
This is a problem in a bioinformatics class, and I believe it shouldn't be too difficult probability-wise, but I've a novice in this area. I think I have about what I need, but I'm very unsure.
...
4
votes
2answers
63 views
A proof on multinomial roots
If $x_1,x_2,...,x_{n-1},x_n$ be the roots of the equation $$1 + x + x^2 + ... + x^n = 0$$ and $y_1,y_2,...,y_{n},y_{n+1}$ be those of equation $$1 + x + x^2 + ... + x^{n+1} = 0$$ show that $$(1-x_1)(1-...
0
votes
1answer
84 views
Expectation of the norm of a Gaussian vector raised to a power
Suppose that $a \in \mathbb{R}^n$ has each of its entries being i.i.d. random variables drawn from a Gaussian distribution with mean zero and variance 1, i.e. $a_i \sim N(0,1), i=1, \cdots, n.$
I am ...
1
vote
0answers
75 views
Simplifying $ \sum_{1\cdot m_1 + 2\cdot m_2 + \cdots + n \cdot m_n = n} \frac{1}{m_1 ! m_2 ! \cdots m_n !} t^{m_1 + \cdots + m_n} $
Does anybody know how to simplify the expression like the following?
$$ \sum_{1\cdot m_1 + 2\cdot m_2 + \cdots + n \cdot m_n = n} \frac{1}{m_1 ! m_2 ! \cdots m_n !} t^{m_1 + \cdots + m_n} $$
This ...
3
votes
2answers
42 views
Find Coefficient of Trinomial Where Term has a Coefficient
Given a problem such as "find the coefficient of $a^2b^6$ for $(a+3b+2)^{10}$," how would I go about doing this?
I know the multinomial theorem, but I'm not sure how to approach this problem given ...
0
votes
1answer
98 views
Expansion of summation of power series raised to a power
I need to simplify the expression written below to get the $ x $ term in its simplest form:
$$
\ E=\left(\sum_{k=1}^b \sum_{n=0}^\infty Z_n(a,k) x^\frac{n+k}{2} \right)^t ,\
$$
where
$$
\ Z_n(a,k)=\...
1
vote
2answers
30 views
Need help finding a sum
I found this problem that I'm not sure how to solve. I would appreciate if anyone could point me to the right direction.
I need to find the following sum:
$\sum_{i+j+k=7} (-1)^i(-1)^j\frac{7!}{i!j!k!...
1
vote
1answer
62 views
Prove that the general norm equation satisfies the triangle inequality for n-dimensions [duplicate]
How do you prove that the general norm equation $\sqrt[p]{\displaystyle\sum_{n=1}^{m} |x_n^p|}$ satisfies the inequality $\sqrt[p]{\displaystyle\sum_{n=1}^{m} |x_n^p|}\leq {\displaystyle\sum_{n=1}^{m} ...
2
votes
2answers
122 views
Simplified $\sum_{n=a_1k_1+a_2k_2+\cdots+a_mk_m}\frac{(a_1+a_1+\cdots+a_m)!}{a_1!\cdot a_2!\cdots a_m!}\prod_{i=1}^m \left(f(k_i)\right)^{a_i}$
I'm studying a function of the form
$$b_n=\sum_{n=a_1k_1+a_2k_2+\cdots+a_mk_m}\frac{(a_1+a_2+\cdots+a_m)!}{a_1!\cdot a_2!\cdots a_m!}\prod_{i=1}^m \left(f(k_i)\right)^{a_i}$$
Where the sum is over ...
2
votes
0answers
45 views
closed-form expression for expected value, $E\left\{X_1\cdots X_k\right\}$ for multinomial distribution
Given a multinomial distribution with parameters $n>0$ where $n$ is an integer and event probabilities $p_i= 1/k$ for $i \in \left\{1, \ldots, k\right\}$.
Next, allow that $\mathbf{N}$ is a ...
0
votes
1answer
30 views
Coefficient of x in a geometric sum raised to the power of n
I have an exam in 6 hours I can't work out how to do these questions. Any help would be greatly appreciated.
a) Compute the coefficient of the term $x^{70}$ in the expansion of the polynomial $(1 + ...
1
vote
2answers
84 views
Number of positive integral solutions in the given inequality
Find the number of positive integral solutions of the inequality $$3x+y+z \leq 30$$
My attempt:
Introducing a dummy variable '$a$' then the equation becomes $3x+y+z+a=30$, where $x,y,z \geq 1$ and $a\...
1
vote
1answer
100 views
Negative multinomial theorem?
Multinomial Theorem:$$(a+b+c\dots)^n=\sum_{i+j+k\dots=n}{n!\over i!j!k!\dots}(a^ib^jc^k\dots)$$
Negative Binomial Theorem:
$${1\over(a+b)^n}=\sum\binom{-n}{k}a^kb^{-n-k}=\sum(-1)^k\binom{n+k-1}{k}a^...
0
votes
2answers
158 views
Proving the Multinomial Theorem--Collapsing Double Sum (Multiple Summation Step)
I am wanting to figure out how to prove the multinomial theorem, but I am stuck on what particular part which is called "collapsing the double sum". How can I show that these two expressions equal one ...
1
vote
3answers
104 views
Find the the coefficient of $\,x^r\,$ in $\,(1+x+x^2)^n$
I want to be able to explicitly write it as $a_r = \dots $
When using multinomial theorem, I'm getting stuck at 2 conditions, but I'm not able to simplify from there.
I wrote $(1+x+x^2)^n =\...
0
votes
5answers
108 views
What is the coefficient of $x^5$ in the expression $(2 + x - x^2)^5$
I am attempting to find the coefficient of $x^5$ in the expression $(2 + x - x^2)^5$ by using the multinomial theorem. I was capable of performing this task only when the number of elements is 2 (...
0
votes
1answer
507 views
Expectation for Trinomial distribution
I am trying to understand the proof of $\mathrm{E}[xy]=n(n-1)p_1p_2$
where x,y have a trinomial distribution with pmf:
$p(x,y) = \frac{n!}{x!y!(n−x−y)!}p_1^xp_2^yp_3^{n−x−y}$
The proof has the ...
0
votes
1answer
42 views
How to show that inequality of a combinatoric series holds true
I am given the following proof:
\begin{align}
&\sum^T_{i=k} \alpha^i (1-\alpha)^{T-i} \binom{T}{i} \\
&=\sum^T_{i=k} \alpha^i \binom{T}{i} \sum^{T-i}_{j=0} (-\alpha)^j \binom{T-i}{j} \\
&=\...
2
votes
1answer
128 views
A multinomial formula?
We know this equality such as multinomial formula:
$$\sum_{l_1 + \cdots +l_k =s} \frac{s!}{l_1!\cdots l_k!}. x_1^{l_1}\cdots x_k^{l_k} = (x_1 +\cdots+ x_k)^s$$
Can we deduce a simular formula or is ...
2
votes
1answer
71 views
Problem in Multinomial Theorem
If $n$ & $k$ are positive integers such that $$n \ge \frac{(k)(k+1)}{2}$$
then the number of solutions $(x_1,x_2,...,x_k)$ such that $x_1 \ge 1 , x_2 \ge 2,..., x_k \ge k$ are all integers ...
2
votes
2answers
515 views
The number of terms in the Multinomial Expansion $(x+\frac{1}{x}+x^2+\frac{1}{x^2})^n$
I am aware that there is a formula to calculate the number of terms in a multinomial expression $(x_1+x_2+x_3+...x_r)^n$, i.e. $^{n+r-1}C_{r-1}$. However, this is in the case when the terms $x_1, x_2, ...
2
votes
4answers
314 views
Multinomial coefficient recurrence formula
I am having trouble with a problem involving multinomial coefficients. I am not seeking an answer directly, just some guidance. The question asks for $$\binom{n} {k_1,k_2,k_3} = \binom{n-1}{k_1-1,k_2,...
2
votes
1answer
142 views
Is there a combinatorial “proof” for this special case of the multinomial theorem?
I am trying to come up with a combinatorial proof of the following fact:
$$
\sum_{k_1+...+k_m=n}{n \choose k_1...k_m}(-1)^{k_2+k_4+...+k_{2l}}= \begin{cases}
0 & \text{if $m=2l$} \\
1 & \text{...
0
votes
1answer
79 views
Probability the winners are split equally among certain underclassmen and upper classmen?
At a certain university, 20% of all students are freshmen, 18% are sophomores, 21% are juniors, and 41% are seniors. As part of a promotion, the university bookstore is running a raffle for which all ...
1
vote
1answer
53 views
Confused at how to obtain this result [equation (2.211) from PRML]
In Pattern Recognition and Machine Learning by Bishop below is a partial reproduction of equation (2.211) on page 115. I'm not seeing how the LHS equals the RHS. What steps would produce this result? ...
0
votes
1answer
23 views
Are weights updated differently in a regression network vs. a classification network?
Are the weight of a neural network updated differently due to back propagation for a classification network vs. a regression network, if so how?..
My concern comes due to the both network uses ...
1
vote
2answers
142 views
Find the coefficient of $a^5b^5c^5d^6$ in $(bcd+acd+abd+abc)^7$
Find the coefficient of $a^5b^5c^5d^6$ in the expansion $(bcd+acd+abd+abc)^7$.
I tried to use multinomial theorem but failed.I can find out the coefficient of $a^5b^5c^5d^6$ in $(a+b+c+d)^21$.But I ...
0
votes
1answer
86 views
Quadratic Multinomial as Double Sum
What's the trick behind the reformulation of the following line? Is this a special case of the multinomial theorem and if so can somebody provide any details?
\begin{equation}
\int (\sum_{j=1}^{N}...
1
vote
1answer
74 views
How many ways can we group 8 balls into groups of size 3?
We have 8 balls, separating them into groups of size 3, we get 2 groups of size 3 and inevitably a group of size 2:
000|000|00
I wanna see in how many ways is it ...
0
votes
0answers
81 views
multinomial distribution for unequal groups?
I am struggling to find a solution to this real life situation:
-Given 4 groups A B C D, with different probabilities of an outcome Z (pAz pBz pCz pDz)
-given that the number of people in each group ...
0
votes
1answer
37 views
An estimate of euclidean norm via multinomial theorem
I have a problem with the following inequality: suppose $k \in \mathbb{N},$ $x \in \mathbb{R}^{n}$ ($n \in \mathbb{N}$), is it true that
$$
(x_1^2 + \ldots + x_n^2)^k \leq C_{k,n}(|x_1|^k + \ldots + |...
1
vote
0answers
11 views
Multipole expansion of point at $z=-d$ vs point at $z=d$
I'm attempting to solve a problem where the solution involves the multipole expansion of a the relative vector between a point at $z=-d, x=y=0$, i.e., $r = -d, \theta = \pi$ and a point at r.
We ...
1
vote
0answers
1k views
Fisher information matrix for multinomial distribution
I am trying to derive the fisher information matrix for multinomial distribution. I know the pmf for it is:
$$f(x_1,x_2,..x_k;n,p_1,p_2,..p_k) = \frac{\Gamma(\sum_ix_i+1)}{\prod_i\Gamma(x_i+1)}\...
