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

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Finding number of integral solutions

I am really getting confused in this question. Number of integral solutions of the equation. $x_1x_2x_3x_4=770$ options- $2^{11}$ $2^{10}$ $4^4$ $5^5$ I attemtemted it by saying that ...
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limit distribution of multinomial distribution with increasing categories

If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know $$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$ where $\bf{X}=(X_1,\ldots,X_k)^T$ and ...
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1answer
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Multinomial identity - guidance needed

I need hints on a direction to proove that $$\displaystyle\prod_{k=1}^{n} {{k+1\choose2}\choose k} ={{n+1\choose2}\choose1,2,3.....,n}$$ Any ideas?
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1answer
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What is the sum of the coefficients in the expansion of $(x+y+w+z)^{20}$

Does the same method used to find sum of the coefficients for a binomial hold here?
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2answers
24 views

Multinomial Expansion Question: when variables appear more than once

Find the coefficient of $x^{12}y^{24}$ in $(x^3 + 2xy^2 +y + 3)^{18}$. I have been working on this problem for a while now and I cannot figure out how to use the multinomial theorem to solve it. I ...
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28 views

Is it possible to solve problems about finding the number of solutions to a linear equation with the multinomial coefficient?

I have an exercise in combinatorics: There are $6$ white balls, $8$ red balls, $4$ yellow balls and $6$ black balls. In how many ways can I pull $6$ balls? Is it possible to solve it with the ...
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1answer
34 views

Multinomial theorem: find the coefficient of $x^3 y^4$ in $(x+2y+3)^{10}$

I have trouble solving this problem: Find the coefficient of $x^3y^4$ in $(x+2y+3)^{10}$ The reason for that I struggle with this problem, is because it has an higher order (10) the $x^3y^4$. ...
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How many summands are there

I have some problem understanding this Exercise/problem. What is summand ? I have searched for it, but nothing concrete came up. Problem: Look at the multinomial theorem. How many summands are there ...
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32 views

Proof with multinomial.

Let $p$ be a prime number. Prove that $p$ divides the multinomial $$\binom {p}{n_1,n_2,\dots, n_k}$$ such that $n_i \neq p$. I tried some approaches but honestly i have no idea what to do.
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1answer
30 views

Maximizing the trinomial coefficient

Let (n; a,b,c) = n!/(a!*b!*c!) In other words, (n; a, b, c) is the trinomial coefficient. I am trying to find the triplets (a,b,c) which maximize this trinomial coefficient. I have determined, by ...
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1answer
73 views

Terms of $(t_1+t_2+…+t_n)^p$?

Who could show me how many terms are in the expansion of $$(t_1+t_2+...+t_n)^p$$ I totally confused myself and I cannot figure it out:-( I was inspired by GDumphart. Thanks so much! It should be the ...
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2answers
106 views

Find the coefficient of $x^{24}$ in $(1 + x + x^2 + x^3 + x^4 + x^5)^8$

I'm not sure how to go about doing this. Do I find the ways to add up to 24 using the exponents with repetition? Is the multinomial theorem useful here? I also have a feeling that generating functions ...
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0answers
37 views

Dirichlet Distribution - the underlying intuition.

I'm not a math expert, but I need dealing with some math tools for natural language processing research. One of the most common tools is the Dirichlet distribution. I know that with a multinomial ...
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2answers
63 views

using multinomial theorem to expand ($2x -y + 3z)^3$

I know how to set it up and there is for this example $10$ terms. But what is the best way to find the expanded work plus finding the final answer? Sorry if there is a duplicate to this problem. I ...
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1answer
50 views

Combinatorics problem involving multinomial theorem

here is my problem: 4 digits are picked randomly (from 0 to 9), in how many combinations can you have only 2 different digits? Meaning that I would have let us say 2,2,4,4 or 5,5,5,8. So the ...
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53 views

Permutations of a Multi-Set

Find the number of permutations of the multi-set {m.1,n.2}, where m,n $\in N $, which must contain m 1's. I thought the permutation is $\frac{(m+n)!}{m!n!}$ since multi-set is basically a collection ...
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1answer
72 views

Difficulty finding multinomial Coefficients

Find the coefficient of $x^{18}$ in the expansion of $(1+x^{3}+x^{5}+x^{7})^{100}$ I got the following answer: $\binom{100}{96,2,1,1} + \binom{100}{94,6,0,0}$ But the answer given is: ...
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81 views

Conditional probability in multinomial distribution

Consider a multinomial distribution with $r$ different outcomes, where the $i$th outcome having the probability $p_i$, $i$=1,...,$r$, $\sum_{i=1}^r p_i = 1$. Denote $X_i$ be the number of times the ...
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4answers
43 views

dealing cards probability

If a standard deck of cards is deal to 4 players, 13 cards each, how many possibilities are there assuming that it matters which player gets but card order does not matter. Why is the answer not (52 ...
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Challenge: How to prove this identity between bi- and trinomial coefficients?

This question is the continuation of its predecessor. Using the convention that trinomial coefficients $$ \binom{n}{k_1,k_2,k_3}=\frac{n!}{k_1! k_2! k_3!} $$ are zero if $k_i<0$ or $\sum_i k_i\neq ...
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1answer
42 views

Finding number of ways of distributing toys without generating function

Suppose I want to distribute $30$ toys in $30$ boxes. Any number of toys (from the given toys) can be kept in any box. In how many ways can this be done? I know how to solve this problem using ...
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1answer
59 views

Probability With Buckets And Balls

There are 5 non identical balls and 5 non identical buckets. You can place any amount of balls in a single bucket. a) In how many cases are there precisely one bucket empty? b) In how many cases are ...
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1answer
60 views

Proof that ordinary multinomial coefficients rise monotonically to a maximum and then decrease monotonically

While most computations of ordinary multinomial coefficients for the following case require recursive summations, I found here a closed-form solution: $$(1+x+x^2+\cdots+x^q)^L = \sum_{a \geq 0} ...
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1answer
82 views

Is there a fast, reasonably accurate estimator for multinomial PDF?

I am working on a balls in boxes kind of problem, where the probability of a ball ending up in a certain box varies by box, that is, each box has some probability P of getting any ball, all together ...
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2answers
52 views

How to compute the coefficient of an equation?

What is the coefficient of $x^2y^2z^3$ in $(x + 2 y + z)^7 $? This is the question at a test and the correct answer is given as 840. Isn't it $7!/(2!2!3!)$ ?
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1answer
39 views

Question on finding the value of x

If the coefficient of $x^2$ in the expansion of $(k+ \frac 1 3 x)^5$ is $30$. What is the value of the constant $k$?
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87 views

Help needed to derive combinatorics formula.

I am having troubles understanding a combinatorics formula. I would appreciate any ideas or hints, leading to an explanation how this formula might be derived. I came across the formula reading a book ...
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1answer
54 views

Finding the coefficient using the multinomial theorem?

Set $F := F (X, Y, Z) = (X^2 + 3Y − Z^2)^8$. Determine the coefficients with which the following terms appear in $F$. $X^4 Y^2 Z^2.$ $X^{10} Y^2 Z^2$. I would know how to find the coefficient if ...
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0answers
30 views

Coefficients of an infinite polynomial?

$$\left(\sum_{k=1}^\infty \left(\sum_{n=1}^\infty a_n x^n\right)^k (-1)^{k+1}\right)^2 =\sum_{j=1}^\infty b_j x^j$$ How to write $b_{n}$ with $a_{n}$?
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Special case coefficient sum in multinomial equation

I need to find the sum of coefficients of $x^{c}$ in the general equation $(1+x)^{a_1}(1+x^2)^{a_2}...(1+x^m)^{a_m}$, where $c$ is a multiple of $m+1$. For example in $(1+x)^2(1+x^2)$ the ...
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1answer
42 views

For $(1+x+x^2)^n = A_0 + A_1x + … + A_{2n}x^{2n}$, prove that $(n-r)A_r + (2n -r+1)A_{r-1} = (r+1)A_{r+1}$

My try: One way to do this: Differentiate the original expression Divide the resultant expression with the original expression Compare coefficients of $A_r$ on both sides This will ...
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0answers
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Multivariate Taylor Polynomial

The Exercise: Calculate the Taylor polynomial of degree 3 of $f(x,y,z)=x^5y^4z^3$ at $(1,1,1)$ in an arbitrary direction $h$. Use Taylor's theorem to get a bound on the remainder when using this ...
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Help to understand and apply the multinomial theorem

I am reading about the multinomial theorem here How do I read the summation notation in this line: Also, can someone please show me how to apply it to the following expansion: $$\left( ...
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2answers
349 views

Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
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4answers
204 views

Coefficient of $x^n$ in the series

How will we find the coefficient of $x^n$ in the following series: $$(1+x+2x^2+3x^3+...)^n$$ Please suggest if there is some formula or if it can be computed using the computer in $\log n$ time. I ...
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1answer
53 views

Unordered multinomial coefficients

Let $\{n_1,\ldots,n_k\}$ be a partition of the integer $m$, that is $m=n_1+\ldots+n_k$, and denote by $\mathcal{P}_m$ the set of all such partitions. For a partition $\pi\in\mathcal{P}_m$, the ...
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Multinomial distribution — sum of squared probabilities

Suppose we have an unbiased multinomial random variable with $n$ trials and $k$ categories. So $X_1, \dots, X_k$ sum to $n$ and have jointly a multinomial distribution with probability $1/k, \dots, ...
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168 views

Expectation value of certain number of trials of multinomial distribution.

Player can extract card from deck (the size of deck is infinite) to obtain one of $k$ kinds of cards, and the possibility of obtaining each kind is given by $p_i$. (Obviously $\sum_{i=1}^{k} p_{i} = ...
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3answers
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Find the Coefficient of $x^{25}$ in…

Find the coefficient of $x^{25}$ in $(1+x^3+x^8)^{10}$ using ordinary generating functions? Could someone help me figure out this problem using generating functions? My initial thought was to using a ...
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1answer
72 views

Prove Multinomial Coefficient (Probability Theory)

Prove that the multinomial coefficient given by: $$ \binom{n}{n_1}\binom{n-n_1}{n_2}\binom{n-n_1-n_2}{n_3}\cdots\binom{n-n_1-n_2-\dots-n_{k-1}}{n_k} $$ equals the following expression $$ ...
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2answers
201 views

Find the coefficient of a power of x in the product of polynomials - Link with Combinations?

I came across a new set of problems while studying combinatorics which involves restrictions to several variables and use of multinomial theoram to evaluate the number of possible combinations of the ...
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183 views

A bin and balls problem

Throw 7 balls into 7 bins. Given there are exactly 2 empty boxes, find the probability that 1 bin contains 3 balls, and thus the other 4 bins contain 1 ball each. I know I could use a multinomial ...
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Combinatorics And Counting

A class consists of 3 boys and 6 girls willing to form 3 groups of 3 called Groups A, B, C. How many ways are there to assign 9 of them to Groups A, B C? I started with $\frac{9!}{3!3!3!}$ but it ...
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1answer
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How to expense $(a+b)^\alpha$ into multinomial with $\alpha \in \mathbb{R}$?

As we all know, the binomial expension is as follows $$ (a+b)^2 = a^2 +2ab +b^2. $$ When the power number is a real number, not a integral, how to expense $(a+b)^\alpha$ into multinomial with $\alpha ...
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1answer
37 views

Are values of multinomials distinct for distinct sets of integer partitions in the denominator?

Let a multinomial be denoted by $$M(n, K) = {n! \over {\prod k_j!}}$$ where $K= (k_1, k_2, ..., k_n)$ and $k_1 \ge k_2 \ge ... \ge k_n$. It is obvious that K is an integer partition of n. Then, my ...
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Proof of multinomial theorem.

I have this proof of multinomial theorem by induction from the Instructor's Solution Manual for Probability and Statistics, 3rd Ed. by DeGroot and Schervish (Addison Wesley/Pearson, 2002). The proof ...
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How could I describe a function whose domain is x>=1 for integers, starts at 3 f(1)=3, then multiplied by 2 f(2)=6, then by 3 f(3)=18, repeat

$$f(1)=3 \quad f(2)=6\quad f(3)=18\quad f(4)=36 \quad f(5)=108$$ How can I define this function? The function is recursive and multiplies by 2 then 3 alternatively. I know I could solve this in ...
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1answer
44 views

The multinomial formula as three Pochhammer rising factorials

I need to describe: $${n \choose k,0,l,0,m}$$ as three rising factorials. How can I do this? As far as I know I can delete zero's, so it would be: $${n \choose k,l,m}=\frac{n!}{k!l!m!},$$ where ...
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How many different words can be formed using all the letters of the word GOOGOLPLEX?

How many different words can be formed using all the letters of the word GOOGOLPLEX? I tried answering this problem and came up with the formula $n!/a!b!c!$ where ...
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2answers
165 views

Combinatorical proof regarding multinomial coefficients

Okay, so combinatorics isn't exactly my strong suit, so bear with me. I'm asked to prove the following combinatorically: If n is a nonnegative integer and k is an integer, then $$\sum_j {n \choose ...