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Questions tagged [multinomial-coefficients]

For questions related to multinomial coefficients, a generalization of binomial coefficients.

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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 ...
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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 ...
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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}) ...
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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 ...
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2answers
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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 ...
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3answers
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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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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\...
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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 ...
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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 $...
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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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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) ...
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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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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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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
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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)]\...
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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} ...
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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 $n= 2,3,4,5,6,7,8...$ ...
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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 ...
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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 ...
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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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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 \...
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2answers
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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/...
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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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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 ...
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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
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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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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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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)=\...
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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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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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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 ...
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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 + ...
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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 ...
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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 $i,j,k$...
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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 ...
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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: ...
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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\...
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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 ...
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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. ...
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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=...
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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! \...