Questions tagged [multinomial-coefficients]
For questions related to multinomial coefficients, a generalization of binomial coefficients.
0
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0answers
16 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
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0answers
24 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
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1answer
28 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}) ...
8
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1answer
109 views
Unexpected proof that $\alpha!$ divides $k!$ if $\alpha_1 + \dots + \alpha_n = k$.
Let $\alpha = (\alpha_1,\dots, \alpha_n) \in \mathbb{N}_0^n$ be a multiindex with $\alpha_1 + \dots + \alpha_n = k$. Let $\alpha! = \alpha_1! \dots \alpha_n!$ with the convention that $0! = 1$.
I ...
0
votes
2answers
52 views
How many passwords with 8 characters have :
How many passwords with $8$ characters ( $26$ lowercase $26$ uppercase and $10$ digits ) have :
a) exactly $3$ lowercase characters $3$ uppercase characters and $2$ digits
b) all characters are ...
1
vote
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 ...
2
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1answer
33 views
An urn contains $4$ red balls, $6$ green balls and $8$ blue balls. Probability with versus without putting the balls back
If we had to determine the probability of getting $3$ red balls $2$ green balls and $4$ blue balls after picking a ball, noting its color and putting it back it would be ${9\choose 3,2,4}\cdot({4\...
1
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0answers
41 views
coefficients in the expansion of multivariable expression
Consider the expansion of the following $N$ variable expression
$$ D_N(z_1,\ldots,z_N)=\prod_{1\leq j<k\leq N}\left(1-\frac{z_j}{z_k}\right)\left( 1-\frac{z_k}{z_j} \right) $$
For example, in the ...
1
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1answer
118 views
Prove that $\|\mathbf{T}^n\|^2=\sum_{|\alpha|=n}\frac{n!}{\alpha!}\|\mathbf{T}^{\alpha}\|^2.$
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.
For ${\bf A} = (A_1,...,A_d) \in \mathcal{L}(E)^d$, the norm of ${\bf A}$ is given by
$...
1
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1answer
239 views
Proof explanation related to multinomial coefficients
Let $\mathbf{X}= (X_1,\cdots,X_d)$ be a $d$-variables and $\mathbf{G}(n,d)$ denotes the set of all functions from $\{1,\cdots,n\}$ into $\{1,\cdots,d\}$.
If the variables $X_k$ are commuting i.e. ...
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0answers
15 views
Confidence interval for multinomial parameters
I wonder how we can make confidence intervals for the parameters of a multinomial distribution. For example, suppose that we ask in a survey: what's your favourite hot drink: 1) Coffee 2) Thee 3) ...
3
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1answer
39 views
Solution using multinomial theorem raised to a negative power
The number of ways of selecting exactly $ 4 $ fruits out of $ 4 $ apples, $ 5 $ mangoes, $ 6 $ oranges is...
A) $ 10 $
B) $ 15 $
C) $ 20 $
D) $ 25 $
I did the solution writing all the possible ...
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0answers
33 views
Runtime of computing the coefficient of a product of multinomials?
Suppose I have $k$ variables, $ x_1, x_2, ... x_k $ and $m$ expressions in the form $ (1 +$ the product of some subset of $x_1 ... x_k)$ – for instance, $(1 + x_1)$ or $(1 + x_1x_2x_5)$ could be one ...
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1answer
24 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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1answer
24 views
Is $E[g(x_i)g(x_j)]=E[g(x_i)]\,E[g(x_j)]$, for $x_i$ multinomial?
Consider the multinomial distribution (Wikipedia):
and let $g:\mathbb{R}\longrightarrow \mathbb{R}^+$ a smooth function. I would like to know if one can show the identity
$$E[g(x_i)g(x_j)]=E[g(x_i)]\...
2
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1answer
135 views
Proof of a Binomial expression summation
Let $x,y$ be probabilities and $n$ is some integer. Show that:
$\displaystyle \sum_{n_0=1}^n \sum_{m=0}^{min(n_0-1,n-n_0-1)} \binom{n_0-1}{m}\binom{n-n_0-1}{m}x^m(1-x)^{n_0-m-1}y^m(1-y)^{n-n_0-m-1} ...
3
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1answer
105 views
How to expand product of $n$ factors.
I have a product say
\begin{equation}
F(a,n,x) = \prod _{j=0}^{n}(1-{a}^{n-2\,j}x)
\end{equation}
I want to expand and hope to have general terms of the coefficients. I did for $n= 2,3,4,5,6,7,8...$ ...
3
votes
1answer
58 views
Counting ways to divide $10$ kids into $2$ teams of $5$. Why divide by $2!\,$?
Here's the question:
In order to play a game of basketball, $10$ kids at a playground divide themselves into two teams of $5$ each. How many different divisions are possible?
The solution given ...
0
votes
1answer
43 views
Upper bound for the sum $\left|\sum_{i=1}^N x_i\right|^p$
Since it is a simple looking question, this might be asked before (however I was not able to find it).
I am looking for a bound in the form,
$$\left|\sum_{i=1}^N x_i\right|^p \leq C(p,N) \sum_{i=1}^N ...
1
vote
0answers
114 views
Sum of multinomial coefficients
It is well-known (using for example the Vandermonde's convolution identity) that
$$\sum\limits_{j=0}^n{n \choose j}^2={2n \choose n}.$$
During my calculation I got the following sum
$$\sum\limits_{k_1+...
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votes
1answer
57 views
Expansion of the cube of the sum of N numbers
I know that the expansion of the square of a summation can be expressed as:
$$\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$$
where $a_n \in \...
3
votes
2answers
64 views
How can one calculate the distribution of this “multinomial” analog of the geometric distribution?
The specific word problem that motivated this question was:
Generate random numbers 0-9 uniformly. Define $W$ to be the number of trials required for at least one 4, at least one 5, and at least ...
1
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1answer
59 views
Is there a trick to compute this multinomial-looking sum?
The series I want to sum has this form
$\displaystyle \sum_{} 1^{l_1}(1+c)^{l_2} (1+2c)^{l_3} \cdot \ldots \cdot (1+(N-1)c)^{l_{N}}$
for some constant $c$ and positive integers $N$ and $L$. Here ...
1
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0answers
22 views
Maxima and Minima of sinusoidal harmonics with unequal co-efficients
This question is related to a similar question I asked before but this time the equation has changed. Maxima and minima of sinusoidal function of harmonics
The equation to which i am trying to find a ...
0
votes
1answer
47 views
In probability, Is there a relation between k-permutations and the multinomial coefficient ?
If we want to place 8 rooks on an 8x8 chessboard, then the number of all the possible placements is 64!/(64-8)! which is just 64-P-8 (k-permutation)
But, can't the same problem be approached as ...
4
votes
2answers
432 views
Combinatorics with arrangements of the word UNIVERSALLY
How many ways are there to arrange the letters in UNIVERSALLY so that the four vowels appear in two cluster of two consecutive letters with at least 2 consonants between the two clusters?
For this, ...
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1answer
66 views
Question regarding $C(n)$ and $B(n)$ in Hanson's proof that $\prod\limits_{p^a \le n} p < 3^n$
In Denis Hanson's proof, he defines two terms:
(1) $B(n)$: which is the Least Common Multiple of $\{1,2,\dots,n\}$
$$B(n) = \prod\limits_{p^a \le n}p$$
(2) $C(n)$: an integer
$$C(n) = \dfrac{n!}{\...
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1answer
82 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 ...
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0answers
34 views
Question on the final step of Hanson's proof that $\prod\limits_{p^a \le n}p^a < 3^n$
In Denis Hanson's proof that $\prod\limits_{p^a \le n} < 3^n$, I am confused by the final step.
He proceeds to show that:
$$C(n) = \frac{n!}{\lfloor n/a_1\rfloor!\lfloor n/a_2\rfloor!\lfloor n/...
1
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0answers
74 views
Simplifying $ \sum_{1\cdot m_1 + 2\cdot m_2 + \cdots + n \cdot m_n = n} \frac{1}{m_1 ! m_2 ! \cdots m_n !} t^{m_1 + \cdots + m_n} $
Does anybody know how to simplify the expression like the following?
$$ \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} $$
This ...
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votes
1answer
55 views
Distribution of sum of iid Categorical Random Variables with outcomes $-1$, $0$ and $1$
The problem I'm addressing is to find the probability mass function of the sum of $n$ i.i.d. Random Variables, each of them having a categorical distribution with outcomes $-1$, $0$ and $1$ with ...
0
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0answers
24 views
Series representation of $\sin^n(x)$ and also applied to other functions
I'm trying to find a power series representation for a function $f^n$ where $n$ is a natural number. In general, I understand that if I can approximate $f$ with the power series $\sum_{k=0}^\infty ...
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2answers
35 views
Find Coefficient of Trinomial Where Term has a Coefficient
Given a problem such as "find the coefficient of $a^2b^6$ for $(a+3b+2)^{10}$," how would I go about doing this?
I know the multinomial theorem, but I'm not sure how to approach this problem given ...
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2answers
88 views
Sum of coefficients of binomial expansion in special case
I need to find the sum of coefficients in expansion of
$$(x_1+x_2+x_3+x_4+x_5+x_6+x_7)^{11}$$ in which degree of any variable is not zero?
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1answer
89 views
Expansion of summation of power series raised to a power
I need to simplify the expression written below to get the $ x $ term in its simplest form:
$$
\ 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)=\...
2
votes
4answers
109 views
Multinomial Expansion
Given $x_0 \ldots x_k$ and $n$, Define $$f(Q)=\sum_{\substack{n_0+\ldots+n_k=n \\ n_0,\ldots,n_k \ >=0 \\n_1+2*n_2+\ldots+k*n_k=Q}} \binom{n}{n_0,\cdots,n_k}x_{0}^{n_0}\ldots x_{k}^{n_k}$$. Note ...
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votes
0answers
136 views
Adjusting for overcounting
In my book
for this question:
How many ways are there to split a dozen people into 3 teams, where each team has 4 people?
The answer is:
There are $\frac{12!}{4!4!4!}$ ways to divide the ...
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1answer
65 views
Multinomial coefficient formula
I was reading about multinomial coefficient, when found this:
$\frac{N!}{k_1!\cdot k_2!\cdot k_3!}=\frac{N!}{(k_1+k_2)!\cdot k_3!}\cdot\frac{(k_1+k_2)!}{k_1!\cdot k_2!}$
Can someone show and explain ...
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1answer
145 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 ...
2
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0answers
42 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 ...
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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
112 views
Determine the coefficient of $wx^3y^2z^2$ in $(2w -x + y -2z)^8$
They provide a similar example:
Similarly provided example
I tried to set mine up the same way, so I had
My Answer so far:
Can someone let me know if I'm even close? In the example, I have no idea ...
0
votes
0answers
42 views
Multinomial expansion indices change to computable sum
I was wondering if it was possible to expand a trinomial expression
$$
(a+b+c)^3=\sum_{0 \leq i,j,k \leq 3;i+j+k=3}\frac{3!}{i!j!k!}a^i b^j c^k
$$
in terms of simple sums (not just referencing $i,j,k$...
1
vote
1answer
38 views
Probability of $x$ non-zero multinomial coefficients?
Let $\prod_{k=1}^K p_k^{c_k}$ be the joint probability distribution of some $n$-long memoryless random process (thus, $\sum_{k=1}^Kp_k=1$), where each $c_k$ tells how many times the $k$-th element ...
1
vote
2answers
57 views
How many ways can N items be shared by 3 anonymous people
I will use a specific example to demonstrate my question.
Take 9 items, they can be shared by 3 anonymous people in the following ways:
...
1
vote
2answers
80 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
73 views
Additive Analog of Multiple Integrals for Zeta Function
A rather famous formula for the Reimann-Zeta function states:
$$\zeta(S) = \int_{[0,1]^S} \frac{1}{1- \prod \limits_{i=1}^{S}x_i} \ \prod_{i=1}^S \text{d}x_i$$
Now, I understand reasonably well why ...
0
votes
1answer
230 views
Multinomial distribution and covariance
A homework question asks:
Let (X, Y, Z) have a multinomial distribution with parameter n = 3, p1 = 1/6, p2 = 1/2, p3 = 1/3.
Find cov(X, Y).
Hint: first find the joint p.m.f. of X and Y.
...
4
votes
1answer
110 views
Analytic function $f$ such that $(f(z))^n= f(z^n)$
Let $f(z) = \sum_{k\geq 0}a_kz^k$ be an analytic function, where $a_k\in\mathbb{C}$ for $k\geq 0$. I am trying to get some conditions for $a_k$ that give us the general form of $f$ such that $(f(z))^n=...
2
votes
2answers
219 views
Sum of Product of Central Binomial Coefficients
The following constants have appeared in my research and I was wondering if they have a simpler expression. I have computed some values and I don't see any obvious combinatorial answer.
$$
c_n:= n! \...
