# Questions tagged [multinomial-coefficients]

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

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### Trinomial equation - combinatorial explanation

Assume that $e+f+g=n$. Prove: $P(n;e,f,g)=P(n-1;e-1,f,g) + P(n-1;e,f-1,g) + P(n-1;e,f,g-1)$ Where $P(n;e,f,g)$ is the tri-nomial-coefficient of n over e,f,g ($\frac{n!}{e!f!g!}$) Combinatorially.
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### bounding extended binomial coefficients from above

Given natural $i,m\ge 1$, how large can the largest coefficient of the polynomial $$(x^0+x^1+\dots+x^{m-1})^i$$ (viewed as polynomial in $x$) be? A trivial upper bound is $m^i$, perhaps even $m^{i-1}$....
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### Class of 16 Participants Answering 6 Questions in Subsets of 4, Each One in A Different Combination With All Pairings Covered

16 Participants are arranged into 4 groups of 4. The participants work together on a question within their groups. Next the groups are rearranged into another 4 groups of 4, where they work on the ...
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### Number of outcomes in permutation invariant Multinomial Distributions

I have a dice with $K$ many outcomes. I am rolling this dice $n$ times. Assume $k_i$ denotes number of times class $i = 1,\ldots,K$ appears in $n$ many rolls. I am wondering, in how many different ...
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### Assign the number of ways to share 16 identical objects

In how many different ways can be shared 16 identical objects to 7 different persons such that 3 of them can accept maximum of 2 objects, 3 of them at least 2 objects and for the other person don't ...
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### Find the coefficient of $x^k$

Find the coefficient of $x^k$ in $$\frac{1}{(1+x)(2-9x)}.$$ This problem is in chapter of Algebraic Tools, using generating functions.
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### Monotonicity of a probability that is related to a multinomial distribution.

Consider a multinomial distribution with three outcomes. Let $x_i$ denote the number of occurences of the $i^{th}$ outcome, and the $i^{th}$ outcome occurs with probability $p_i$, $i=1,2,3$. Let $n$ ...
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### Intuitive meaning for multinomial coefficient. (Why only one?) [closed]

Introduction I am doing a (university) probability course, wherein there seem to be occasional errors in what is taught. [Edit: this is not just my opinion.] <...
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### A certain composition into the elementary symmetric polynomials

Preliminaries Let $\mathbb{F}$ be a field such that $\operatorname{char}(\mathbb{F})\neq2$. Let $n$ be a non-zero natural number. Let $\mathbb{F}\left[x_1,x_2,\ldots,x_n \right]$ be a polynomial ...
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### Coefficient of $x^i$ in $(x+x^2+…+x^k)^n$

Is there any general way to find coefficient of $x^i$ in $(x+x^2+...+x^k)^n$ It is easy to solve when k is small like $k=3$ or $k=4$ by using multinomial coefficient But how can we solve a problem: ...
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### Prove $\binom{n}{k_1,…,k_m} = \sum_{i=1}^m \binom{n - 1}{k_1,…,k_{i - 1},..,k_m}$

I have a question ask to prove $$\binom{n}{k_1,...,k_m} = \sum_{i= 1}^m \binom{n - 1}{k_1,...,k_{i - 1},..,k_m}$$ I'm not sure how to approach this question, but the only thing that I noticed the LHS ...
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### Given $(x+y+z)^{15}$ find the coefficient of $x^2y^{10}z^{3}$ [duplicate]

Given $(x+y+z)^{15}$ find the coefficient of $x^2y^{10}z^3$ Weak in this chapter. Don't know how to proceed. Please help Sorry for the typo made before
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### The maximal term of the multinomial distribution

(Feller volume 1, Q28, p.171) Suppose that we have the following binomial distribution $$\frac{n!}{k_1!k_2!... k_r!} p_1^{k_1}p_2^{k_2} ... p_r^{k_r}.$$ Prove the theorem. The maximal term of the ...
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### Is there a way to find the coefficients of this polynomial without expanding it?

If the expression $(x^3-x^2y+xy^2+y^3)^3$ is expanded and simplified, what is the sum of all the coefficients of the resulting polynomial? I know the answer is 8, and I think last time I did it I ...
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### counting allowable/admissible paths on a grid with obstacles

I have a grid where some paths are removed. where the following path is admissible, but this path is not. How can I go about finding how many admissible paths are there on the following 2 grids (...
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### calculating total number of allowable paths

I seem to be struggling with the following type of path questions Consider paths starting at $(0, 0)$ with allowable steps (i) from $(x,y)$ to $(x+1,y+2)$, (ii) from $(x,y)$ to $(x+2,y+1)$, (iii)...
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### greatest term in multinomial expansion proof

Can anyone give the proof for greatest term in multinomial expansion (coefficients of x are not 1)and an intuitive sense for the formula.Our book just contains the formula without any understanding of ...
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### calculating allowable paths

I have the following question, usually I'd solve by a modification of pascals triangle but I'm not sure how to approach this using pascals since step D is problematic. How could I go about this? ...
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### Finding a coefficient of $x^{57}$ in a polynomial $(x^2+x^7+x^9)^{20}$

So the task is to find a coefficient of $x^{57}$ in a polynomial $(x^2+x^7+x^9)^{20}$ I was wondering if there is a more intelligible and less exhausting strategy in finding the coefficient, other ...
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### Multinomial Distribution — How to calculate percentiles?

I've read the rules and searched but I do not even know what I'm looking for. Here is my problem: Suppose I have a bag containing three different marbles: red, green, and blue. I am drawing a single ...
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### calculating total number of allowable paths from $(0,0)$ to $(5,5)$

I'm looking at paths starting from $(0,0)$ with the following allowable steps : 1) from $(x,y)$ to $(x,y+1)$ 2) from $(x,y)$ to $(x+1,y)$ 3) from $(x,y)$ to $(x+2,y+1)$ how can I determine the ...
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### How many ways to form a 4 letter word when order doesn't matter and letters need not be different

Can someone please explain this. How many 4 letter words are there, when order doesn't matter and letters can be repeated ? IF I do in one approach I get $\frac{26^4}{4!}$ (26 letters for each ...
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### In the multinomial expansion of $(a+b+c)^5$ , why does “a” have $5 \choose 1$ positions? [closed]

Also, why does b has $4\choose 2$ positions and c has $3 \choose 2$ positions? , i.e. $(a+b+c)^5=a^5+...+abbcc+babcc+cabcb+....$
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### What is the math behind this Python program generating multinomial coefficients?

I wrote a Python program that is using recursion to generate multinomial coefficients - see next section. Mathematically it is also using recursion by 'decrementing down to the boundary'. My ...
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### Combinatorics and Multinomials: How to find the number of distinct terms?

How do I find the number of distinct terms in (a - 2b + 3c + d)ⁿ where n = 17? I started the problem by realizing that the multinomial expansion says that to find the number of distinct terms for (a + ...
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### Approximation of multinomial distribution for large N (Expected end-to-end distance of a random walk on a hypercubic lattice in arbitrary dimension)

I was trying to derive an expression for the expected end-to-end distance of a random walk on a hypercubic lattice with sides of length $l$ in an arbitrary dimension $d$ after $N$ steps. I found that ...
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### Multivariable Linear Regression Coefficient

I have a dataset something like this: ...
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### Coefficient shortcut

I'm taking the product of a specific set of polynomials of idempotent variables and want to find a shortcut for counting groups of homogeneous coefficients without performing all the calculations. ...
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### Can a binominal (or multinomial) coefficient be computed efficiently?

It would seem that a preceding query would be on point, but not really for me. One of the answers comes close, but it isn't complete as is. Since the answer is going to be an integer, all the ...