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

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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
58 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
28 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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36 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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What's the difference between multinomial coefficients and the number of weak compositions? [closed]

Multinomial coefficients and the number of weak compositions deal with $n$ balls in $k$ boxes so what's the difference?
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81 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
26 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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27 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
41 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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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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62 views

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
190 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
187 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
44 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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45 views

Sum of multinomial coefficients

I have proven the multinomial theorem and the wikipedia page says we can deduce that $$\sum_{x_1+x_2+...+x_k =n}\binom{n}{x_1,x_2,...,x_k} = k^{n}$$ but I cannot see how it follows as a result. ...
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72 views

How far can the binomial theorem be stretched?

Good afternoon, The multinomial theorem generalizes the binomial theorem. By some reason Wikipedia leaves out upper limits of the sums, and I don't trust sums without clear limits (nor that article, ...
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1answer
144 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
103 views

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
55 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
130 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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147 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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55 views

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
48 views

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
34 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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255 views

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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2answers
73 views

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
40 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
143 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 ...
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102 views

Divisibility of multinomial by a prime number

What is the condition for divisibility of multinomial $ \dbinom {n}{x_1, x_2, \dots, x_k} $ by a prime $p$? Update: I tried to solve using a generalisation of Lucas Theorem by representing the $n$ ...
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What are the coefficients of the polynomial inductively defined as $f_1=(x-2)^2\,\,\,;\,\,\,f_{n+1}=(f_n-2)^2$?

Let $\{f_n(x)\}_{n\in \Bbb N}$ be a sequence of polynomials defined inductively as $$\begin{matrix} f_1(x) & = & (x-2)^2 & \\ f_{n+1}(x)& = & (f_n(x)-2)^2, &\text{ for all ...
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603 views

Probability: Biased Die

Suppose we have three 6-sided die that all share the same common bias: For a single dice: let the probability of rolling a 2 or $P(2) = 2{\times}P(1$), let the probability of rolling a 3 or $P(3) = ...
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1answer
49 views

Triangle of Multinomial Coefficients

What is the "Triangle Of Multinomial Coefficients" seen here: http://oeis.org/A036038 (OEIS: A036038) I can see that the diagonals of this triangle are just factorials... for example the last number ...
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1answer
48 views

(CHECK) Cardinality of Terms in the Expansion of a Product of Multinomials

QUESTION: How many terms are there in the expansion of $$(x+y)(a+b+c)(e+f+g)(h+i)$$ I'd like some help with this one, but I'd also like to discuss a method of generalization on the problem, ...
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1answer
59 views

Is it possible to “customize” the multinomial distribution to your specifications?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
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450 views

Find the coefficient of $x^{20}$ in $(x^{1}+⋯+x^{6} )^{10}$

I'm trying to find the coefficient of $x^{20}$ in $$(x^{1}+⋯+x^{6} )^{10}$$ So I did this : $$\frac {1-x^{m+1}} {1-x} = 1+x+x^2+⋯+x^{m}$$ $$(x^1+⋯+x^6 )=x(1+x+⋯+x^5 ) = \frac {x(1-x^6 )} {1-x} = ...
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1answer
145 views

How to calculate probability using multinomial distribution?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
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1answer
201 views

Find coefficient of $x^{100}$ in the power series expansion of $\frac{1}{(1-x)(1-2x)(1-3x)}$

I'm trying to find to coefficient of $x^{100}$ of $$\sum_{n=0}^{∞}a_n x^{n}\ =\frac{1}{(1-x)(1-2x)(1-3x)}.$$ I used the sum: $$\frac{1}{1-x}\ = 1+x+x^2+\ldots.$$ So : $$\frac{1}{(1-x)(1-2x)(1-3x)}= ...
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66 views

Math Formula for Rank of Elements in Given Order

Say we have a string with repeated letters, 'abcaacb', and we want to determine it's rank alphabetically among other strings that contain the same exact letters. So rank 1 would be 'aaabbcc', and the ...
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1answer
119 views

Find Fourier Series coefficients of x=1 line function.

I want to know that can we find the Fourier series coefficients of the periodic signal x=1 where the limits are from -infinity to +infinity. The problem arises with the limits and it will converge to ...
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1answer
98 views

Expression for sum of $k$-products of $n$ variables

Given $n$ variables there are $n \choose k$ different terms that are the product of $k$ different variables. For example, in the case that $n = 3$, the $k$-products of the variables $x_1, x_2, x_3$, ...
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237 views

Different Perspectives of Multinomial Theorem & Partitions

There are 2 important interpretations of the multinomial theorem and coefficients. 1: Determining the number of ordered strings that can be formed using a set of letters. For example, with 1 m, 4 ...
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827 views

Expand Trinomial Raised to an Exponent, (x+y+z)^4, with Multinomial Theorem

Use the multinomial theorem to expand $(x+y+z)^4$. To calculate the number of terms, you apply the following formula: $\binom{n+r-1}{n}$. Here, n=4 and r=3. So $\binom{6}{4}=15$. I don't ...
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1answer
1k views

Multinomial Theorem Example Questions

I'm learning about the multinomial theorem and working 2 examples in a book. I thought I understood the examples until I did example 5c. I don't understand why these two examples are different. In ...
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1answer
149 views

Prove a sum involving multinomial coefficients

I need to prove that if n and m are positive integers, then $$ \sum_{k_1+...+k_m}\binom{n}{k_1, ..., k_m}(-1)^{k_2+k_4+...+k_{2l}}$$ is equal to 0 if m =2l, and is equal to 1 if m = 2l + 1.
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256 views

How many solutions are possible to this equation?

Given $$A+2B+3C=N $$ where $N$ is a given positive integer. $A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$. How many solutions will be there to this equation?
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1answer
94 views

A question concerning multi-indices

I am having difficulties understanding the following formula : $$(x_1+\cdots+x_n)^k=\sum_{\alpha,|\alpha|=k}\frac{|\alpha|!}{\alpha!}x^\alpha $$ where $\alpha$ is a multi-index. I find this notation ...
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38 views

How do i apply multinomial laws in this question?

the Question is i assume i have 15 students in class A grade obtain probaiblity = 0.3 B grade obtain probability =0.4 C grade obtain probability = 0.3 and I have this question What is the ...