Questions tagged [multinomial-coefficients]

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

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32 views

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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80 views

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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22 views

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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0answers
32 views

Trinomial Expansion and Complex Numbers

Consider the expression $$(1 + x + x^2)^n = A_0 + A_1x + A_2x^2+\cdots + A_{2n-1}x^{2n-1} + A_{2n}x^{2n}$$ (where $n$ belongs to positive integers). $n\equiv 0\mod 4 \implies \quad A_0 - A_2 + A_4 - ...
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1answer
525 views

Trinomial Theorem and Complex numbers

Consider the expression $$(1 + x + x^2)^n = C_0 + C_1x + C_2x^2+\cdots + C_{2n-1}x^{2n-1} + C_{2n}x^{2n}$$ (where $n$ belongs to positive integers),then the value of $C_0 + C_3 + C_6 + C_9 + C_{12}+\...
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21 views

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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1answer
19 views

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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76 views

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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Polynomials and mulinomial coefficients [duplicate]

Let it be $$(1+x+x^2+x^3+x^4)^{496}=a_0+a_1x+a_2x^2+...+a_{1984}x^{1984}.$$ Find $$\gcd(a_3,a_8,...,a_{1983})$$ I know the Leibniz Theorem, so all coefficients are of the form $$\frac{496!}{a!b!c!d!e!}...
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Trinomial Summation Equation [duplicate]

I was looking at how to find trinomial coefficients with code and I was puzzled with this equation. $$\binom n k _2 = \binom{n}{-k}_2$$ I know that (nCk) represents n choose k but what does the 2 at ...
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34 views

Polynomials and Number Theory

Given the equation $(1+x+x^2+x^3+x^4)^{496}=a_0+a_1x+a_2x^2+...+a_{1984}x^{1984}$ a) What is the gcd of $(a_1,a_2,a_3,... a_{1983})$? b) Show that $10^{340}< a_{992}< 10^{347}$. By the ...
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2answers
42 views

Multinomial Coefficient of $x^{1397}$ in expansion of $(x^3+x^4+x^5+…)^6$

I have the following problem: Find the Coefficient of $x^{1397}$ in expansion of $(x^3+x^4+x^5+...)^6$ I know how to solve these kind of questions using Multinomial Theorem but since the polynomial ...
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22 views

Understanding coefficients of related polynomials

Im trying to understand this concept in general but as an example lets say I have a polynomial like: $$f=(1+x+x^2)(1+x+2+(x+2)^2+(x+2)^3+(x+2)^4)(1+x+6+(x+6)^2 $$ which can simplify to: $$ f= (x^8 + ...
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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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2answers
55 views

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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1answer
50 views

Sum of multinomial coefficients with bounded indices

I want to know if there's a closed formula for $$ f_{k}(n_{1}, \dots, n_{k}) = \sum_{i_{1} = 0}^{n_{1}} \cdots \sum_{i_{k} = 0}^{n_{k}} \frac{(i_{1} + \cdots + i_{k})!}{i_{1}!\cdots i_{k}!} $$ for $k\...
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2answers
31 views

Binomial coefficient after expansion

I am trying to solve an exercise where in the final step, I need to find the coefficient of $x^7y^5$ in $(x+y)^{12} + 7(x^2+y^2)^6 + 2(x^3+y^3)^4 + 2(x^4+y^4)^3 + 2(x^6+y^6)^2 + 4(x^{12}+y^{12}) + 6(x+...
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1answer
137 views

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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2answers
120 views

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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1answer
39 views

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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2answers
57 views

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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1answer
52 views

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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16 views

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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2answers
99 views

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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63 views

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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1answer
31 views

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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4answers
74 views

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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2answers
84 views

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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3answers
79 views

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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2answers
69 views

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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2answers
68 views

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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1answer
87 views

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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1answer
33 views

Algebraic identity for summing all arrangements of n choose k terms

As the title suggests, I am looking for this type of identity, where k is some integer between 3 and n-2. I know, for example, that $$\frac{(x_1^2+x_2^2+...+x_n^2)-x_1^2-x_2^2-...-x_n^2}{2}=x_1x_2+...
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4answers
121 views

Sum of coefficients of $x^i$ (Multinomial theorem application)

A polynomial in $x$ is defined by $$a_0+a_1x+a_2x^2+ \cdots + a_{2n}x^{2n}=(x+2x^2+ \cdots +nx^n)^2.$$ Show that the sum of all $a_i$, for $i\in\{n+1,n+2, \ldots , 2n\}$, is $$ \frac {n(n+1)(5n^...
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48 views

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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16 views

Multivariable Linear Regression Coefficient

I have a dataset something like this: ...
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1answer
68 views

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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2answers
94 views

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 ...
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2answers
79 views

What's the coefficent of $x^{20}$ in $(x^3+x^4+x^5+…)^5$? Only hint is needed.

I know about Binomial/Multinomial expansion but I got stuck on this series; it doesn't look like anything I've solved before. I already searched for any hint/formula and couldn't find one, any help is ...
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1answer
21 views

A Conceptual doubt on Multinomial Theorem.

I was solving some questions of Multinomial Theorem from Higher Algebra by Hall and Knight when I encountered this question. $\text{If }\left(1+x+x^{2}+\ldots+x^{p}\right)^{n}=a_{0}+a_{1} x+a_{2} x^...
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2answers
35 views

A question on the multinomial expansion

Given that $$(2x^2+3x+4)^{10}=\sum_{i=0}^{20} a_{i}x^{i} $$ Calculate the value of $\frac {a_{7}}{a_{13}} $. I have manually taken all the cases of formation of $x^7$ and $x^{13}$ and arrived at the ...
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1answer
27 views

multinomial sum upper bound

I am looking for an upper bound on the following sum, where $p_1,\dots,p_r > 0$ and $\sum_i p_i\leq 1$. $$ \sum_{0\leq i_1,\dots,i_r\leq n} \binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots ...
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4answers
71 views

Find coefficient in cubic polynomial

We have $$ p(x) = ax^3 + bx^2 + cx +d$$ where $a, b, c, d$ are complex coefficients. We have to find all posible coefficients for: $$ p(1) = 2$$ $$ p(i) = i$$ $$ p(-1) = 0$$ Unfortunately I dont know ...
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1answer
30 views

How to compute the coefficients (ODE using power series method)?

How to Solve $xy'+2y=4x^2;y(1)=5$ using power series method? I've assumed $y = \sum_{m=0}^\infty a_mx^m\implies y' = \sum_{m=1}^\infty m a_mx^{m-1}$. Putting these values in equation in $xy'+2y=4x^...
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3answers
94 views

Find coefficient for $x^{10}$ in $x^3(x^2-3x^3-1)^6$

I'm trying to solve the following problem: Find the coefficient for $x^{10}$ in $x^3(x^2-3x^3-1)^6$. Can I use the multinomial theorem to solve it? I'm unsure how to start.. Thanks!
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1answer
41 views

How to prove that ${n \choose r_{1},\ldots,r_{k}}=\sum_{j=1}^{k}{n-1 \choose r_{1},\ldots,r_{j}-1,\ldots,r_{k}}$ using algebra?

Im trying to prove using algebra that $$ {n \choose r_{1},\ldots,r_{k}}=\sum_{j=1}^{k}{n-1 \choose r_{1},\ldots,r_{j}-1,\ldots,r_{k}} $$ Attempt: The multinomial theorem states that $$ \left(x_{1}+\...
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2answers
186 views

Counting paths in multidimensional grids

Let's consider a grid in a multidimensional space. Each grid has 'n' number of lines, spaced one unit apart. A path (or walk) of 'm' steps is between two grid intersections. A step is a single unit ...
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1answer
41 views

Coefficient of $x^4y^3z^3$ in the expansion of $(5x+y-4z)^{10}$

The coefficient of $x^6y^4$ in the expansion of $(2x-3y)^{10}$ is $$_{10}C_6 \cdot 2^6 \cdot (-3)^4$$ and as for the coefficient of $x^3y^4z^8$ in the expansion of $(x+y+z)^{15}$ is $$_{15}C_3 \...

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