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

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2answers
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Rolling two dice… [closed]

Let $A_n$ be the number of fives, $B_n$ the number of sixes and $C_n$ the number of eights in $n$ rolls of two dices. For which n do we have: $E(A_n) < E(min(B_n,C_n))$ ?
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1answer
44 views

Sum of variances of multinomial distribution.

I've k fair coins, and I would like to know the number of heads obtained in $n$ trials. But that is simple binomial distribution. But if I want to find out how much it varies from binomial ...
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0answers
57 views

Multinomial Theorem for Negative Exponents

Using an analog to Newton's binomial theorem with negative exponents, is it true that $$ \begin{align} \left(\sum_{k=0}^mx^k\right)^{-n} & = \sum_{0\le ...
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1answer
16 views

Numerically stable calculation of multinomial probabilities

I'm looking for a numerically stable method to compute expressions of the form $$\frac{(a+b+c+d)!}{a!b!c!d!}\left(\frac{1}{4}\right)^{a+b+c+d}$$ So far I've been using a compensated sum algorithm to ...
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1answer
44 views

Probability Generating Function of a Negative Multinomial Distribution

Derive the probability generating function (pfg) of a negative multinomial distribution with parameters $(k; p_{0}, p_{1}, ..., p_{r})$ where the k-th occurrence of the event with the probability ...
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2answers
44 views

I need help in calculating the sum of the coefficients of even powers of $x$ in $(1+x-2x^2)^6$

I need to calculate the sum of coefficients of even powers of $x$ in $$(1+x-2x^2)^6$$ I don't know much about the multinomial theorem, but i know the basics pretty well. I have some ideas of solving ...
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0answers
34 views

coefficients of polynomial and binomial expressions

Let us say we are given a polynomial p(x)=$\sum_k a_k x^k$. In order to find $\sum_k a_k$ we simply need to evaluate p(1), and similarly there are many other tricks. Is there any trick to evaluate ...
2
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2answers
87 views

Find the coefficient of $x^4$ in the expansion of $(1 + 3x + 2x^3)^{12}$?

I have not learnt the multinomial theorem yet, and was trying to approach this using the binomial theorem. I divided the terms as $a$ being $(1+3x)$ and $b$ being $2x^3$. I then used $${12\choose ...
4
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3answers
196 views

Sum of coefficients in an multinomial expression.

Q: The sum of all the coefficients of the terms in the expansion of $(x+y+z+w)^{6}$ which contain $x$ but not $y$ is: What I tried to do was make pairs of two terms and the expand it as a binomial ...
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1answer
31 views

Ways of selecting at most n objects from a set containing k distinct elements where each element can occur any number of times.

I have a box with a maximum capacity of n elements. A state of the box is defined by the elements in it. There is an infinitely large heap which has k distinct elements; each element is available ...
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1answer
41 views

Upper bound to multinomial coefficient sum

I'm currently stuck on what seems like a very trivial problem. I have the following calculation $$ \sum_{k_1+k_2=0}^{n} {n \choose n - k_1 - k_2, k_1, k_2}^2 \le \sum_{k_1+k_2=0}^{n} {n \choose ...
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1answer
75 views

$(1+x+x^2)^{1061}=a_0+a_1+…$ then what is the value of $(1-a_1^{2}+a_2^{2}…)$ in terms of $a_n$

$(1+x+x^2)^{1061}=a_0+a_1x+...+a_{2122}x^{2122}$ then what is the value of $(1-a_1^{2}+a_2^{2}-a_3^{2}...)$ in terms of single $a_n$ ? n lies between 0 and 2122 . how to get in terms on $a_n$ ? ...
4
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1answer
127 views

Complex Analysis proof of multinomial expression

I've recently come across the following identity $$ \displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n} $$ A nice complex analysis proof (by Felix Marin, here) follows as: ...
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1answer
61 views

Multinomial Expansion-Example

Is coefficient of $x^{20}$ in $(1-x+x^2)^{20}$ and $(1+x-x^2)^{20}$ same? Can someone tell me how should i apply multinomial theorem to this problem?
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1answer
50 views

Maximum of trinomial coefficient

I read a proof about the simple random walk in 3 dimensions and couldn't understand the following statement: $$\frac{n!}{k!j!(n-k-j)!}$$ has the maximum when $\ k, j $ and $\ n-k-j$ are as close to ...
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1answer
37 views

Find multiplier of x^6 in (a + bx + cx^2)^4 (Multinomail theorem)

Have a simple problem where I need to find multiplier of $x^6$ in $(2+x+3*x^2)^4$ I tried doing it like this: $ a = 2; b = x; c = 3*x^2$ Using multinomial theorem : Possible combinations that give ...
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1answer
20 views

Cofficient of x in a product

How can I efficiently find the coefficient of $x^m$ in the following product - $\prod\limits_{i=1}^{n-1}(1 - p_i + p_ix)$
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1answer
28 views

Is there a straight forward way to count the number of monomials in a multinomial equation?

The multinomial theorem states that: $$(x_1 + x_2 + \dots + x_m)^n = \sum_{k_1 + k_2 + \dots + k_m = n}{n \choose {k_1, k_2, \dots, k_m}}\prod_{1 \le t \le m} {x_t}^{k_t}$$ The number of monomials ...
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1answer
45 views

Integration of this using a multi-dimensional hypergeometric function

I want to try and potentially use a Dirichlet - Hypergeometric Function in order to compute the following integral. I would appreciate some help as I'm stuck on how to go about this is a ...
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3answers
93 views

Number of distinct terms in the expansion of $\big(x+\frac{1}{x}+x^2+\frac{1}{x^2}\big)^{15}$

Number of distinct terms in the expansion of $\bigg(x+\dfrac{1}{x}+x^2+\dfrac{1}{x^2}\bigg)^{15}$ is equal to ? We can write the above as, $$ \bigg(x+\dfrac{1}{x}+x^2+\dfrac{1}{x^2}\bigg)^{15} = ...
2
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1answer
42 views

Multinomial theorem: Number of elements where all coefficients have even powers..

Consider $(a_1+a_2+ \cdots + a_n)^r$. We know that it has $n^r$ elements. I want to calculate the number of elements, where all $a_i$ coefficiants have even powers, i.e. $(a_1+a_2+ \cdots + a_n)^r = ...
2
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0answers
49 views

Generate list of distinct coefficients

Is it possible to find the (distinct) coefficients of monomials such as $$(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3})^4\cdot(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})\cdot(x_{1}+x_{2}+x_{3}+x_{4})^{2}$$ ...
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0answers
17 views

Quickly finding coefficients of $\left(\sum _{i=1}^t x_i\right){}^{p_2} \sum _{i=1}^t x_i^{p_1}$ for certain exponent combinations

I've been helping someone with the following: For the homogeneous polynomial $\left(\sum _{i=1}^t x_i\right){}^{p_2} \sum _{i=1}^t x_i^{p_1}$ and $\left(p_1|p_2\right)\in \mathbb{Z}\land p_1\geq ...
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1answer
54 views

Use of Multinomial theorem.

I have the next identity which I want to prove. $$(\sum_{j}k_j^2)^{s} = \sum_{b_1+\ldots+ b_n =s} \prod_j k_j^{2b_j}$$ Obviously I need to use the Multinomial theorem, but how to procceed from ...
8
votes
1answer
211 views

Proving that there are at least $n$ primes between $n$ and $n^2$ for $n \ge 6$

I was thinking about Paul Erdos's proof for Bertrand's Postulate and I wondered if the basic argument could be used to show that there are more than $n$ primes between $n$ and $n^2$. Is this approach ...
2
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0answers
37 views

Multinomial problem

Suppose one has a nested table of disitnct primes then the permutations of their products produce dupluicates at certain values. For example, letting the primes $\{a,b\}=\{2, 3\}$, the products of ...
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1answer
35 views

What does the multinomial formula state? (Formula provided)

The statement of the formula is kind of cryptic to me. In particular, I don't understand how the formula relates the probability of an object, designated by the index i, occurring $n_i$ times since ...
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2answers
70 views

Rolling a fair die 18 times, what is the probability of rolling 1,2,3,4,5,6 (each) three times?

The problem statement and solution is below. I understand that each die toss is independent but it doesn't make sense to me why the probability is multiplied by the number of sequences, as in the ...
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2answers
175 views

Find the constant term in the expansion of…

How can we find the term independent of $x$ in the expansion of, ...
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3answers
40 views

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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0answers
18 views

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
42 views

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
46 views

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
48 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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0answers
39 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
52 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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votes
3answers
57 views

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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2answers
36 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
61 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 ...
2
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1answer
81 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
155 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
58 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
74 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
74 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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2answers
70 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
79 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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2answers
382 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
56 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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2answers
136 views

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
48 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 ...