Questions tagged [multinomial-theorem]

An extension to the binomial theorem. It gives the expansion of a multinomial $(x_0,\dots,x_{m-1})^n$.

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Find the Coefficient of $x^8$ by Multinomial Theorem Expansion for $(1+x^2-x^3)^9$

Problem: Find the coefficient of the term $x^8$ for the expansion of $(1+x^2-x^3)^9$ Attempt: By the multinomial theorem: $$(1+x^2-x^3)^9=\sum_{b_1+b_2+b_3=9}{9\choose b_1,b_2,b_3}(1)^{b_1}(x^2)^{b_2}(...
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2 votes
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An equivalent formula for $\sum_{1\le i_1\lt i_2 \dots \lt i_n\le n} a_{i_1} a_{i_2} \dots a_{i_n}$

I know that the following holds: $\sum_{1\le i\lt j\le n} a_j a_k = \frac12\left(\left(\sum a_i\right)^2-\sum a_i^2\right)$ The question is: does some equivalent formula holds for products of $q>...
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How to find the $r$-th term of a trinomial expansion?

For a binomial expansion $(x^a + y^b)^n$ finding the rth term simply meant that the exponent of $y$ should be $b$ times $n$. How about for a multinomial expansion, $(x^a + y^b + z^c)^n$? I see in some ...
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6 votes
5 answers
170 views

Find the $x^n$ coefficient of $(1+x+x^2)^n$

I've tried a bunch of different groupings of the three terms so that I could use the binomial expansion forumula, but I haven't been able to go much further than that. This is an example of what I've ...
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2 votes
1 answer
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Issues understanding the multinomial theorem and its multiindex notation

$$(x_1+x_2+...+x_m)^n=\sum_{(k_1 + k_2 +... +k_m) \ = \ n} {n \choose k_1,k_2...k_m} \prod^m_{t=1}x_t^{k_t}$$ Let's do $(a+b+c)^3$. That means $a =x_1, b =x_2, c=x_3=x_m$. The multiindex below the ...
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How to get multinomial sum without coefficients

We know that a multinomial sum is given by: $(x_1 + x_2 + x_3 + ... + x_m)^n = \sum\limits_{k_1+k_2+k_3+...+k_m=n} {n \choose {k_1, k_2, k_3, ... k_m}} \prod\limits_{t=1}^m x_t^{k_t}$ I want to ...
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3 votes
2 answers
72 views

Number of terms in product of two monomials with common terms

I am trying to find the number of terms in the expression $$(x+y+z)^{20}(w+x+y+z)^2$$. I understand that the number of terms in $(x+y+z)^{20} = \binom{22}{2}$ and the number of terms in $(w+x+y+z)^2 = ...
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Is it possible to determine the number of terms in a multinomial expansion, if all terms are exponents of $x$?

Is it possible to determine the number of terms in a multinomial expansion, if all terms are exponents of $x$? For example, number of terms in the expansion of $(1+x^2+x^4+x^5)^7$ ? Clearly, the ...
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Coefficient of $1$ in the expansion of $\left(1+x+\frac{1}{x}\right)^n$

What is the coefficient of $1$ in the expansion of $(1+x+\frac{1}{x})^n$? In other words, what is the sum of the coefficients of $x^ky^k$ in the expansion of $(1+x+y)^n$? Here, $n$ is a positive ...
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-1 votes
2 answers
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Question about multinomial expansion

The teacher briefly glossed over the multinomial theorem and then dropped this seemingly monstrous homework problem on us: Find the coefficient of $x^{12}$ in the expansion of: $(x^5+x^6+x^7+\ldots )^...
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No. of integral solutions of an equation with upper and lower bound without 'generating the function' method

Question from my book Now, I know how to solve this using 'generating the function' method. But I cant figure out what method is used in the solution of the problem given in the book,especially from ...
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7 votes
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How to show $\begin{pmatrix}1&1 \\ 1&1\end{pmatrix}^n = 2^{n-1}\begin{pmatrix}1&1 \\ 1&1\end{pmatrix}$?

Initial note: I'm interested in the combinatorics aspect of the following problem, not how to proof the relation in general. The idea is to show the following relationship: $$ \begin{pmatrix}1&1 \...
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How to apply Multinomial Theorem here?

I know that if $I_1, \dots, I_n$ are finite index sets and for every $i \in \bigcup_{j=1}^n I_j$ there exists $v_i \in \mathbb{R}$, then holds $$ \prod_{j=1}^n \sum_{i \in I_j} v_i = \sum_{(i_1, \dots,...
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3 votes
1 answer
69 views

Multinomial Co-efficient Summation

If $$ \begin{equation} (1+x+x^2+...+x^p)^n=a_0+a_1x+a_2x^2+...+a_{np}x^{np} \end{equation}\label{given}\tag{1} $$ Prove that $$ \begin{equation} S=a_1+2a_2+3a_3+...+np.a_{np}=\frac{1}{2}np(1+p)^n \...
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1 vote
1 answer
38 views

Constrained Sum of Factorials

Consider the sum $$S=\sum_{\substack{r_i>0,\\1\le i\le m\\ r_1+...+r_m=n}}\frac{1}{r_1!...r_m!},$$ where $m,n$ are fixed, positive integers, and the $r_i$ are integers. If there were no ...
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Multinomial Theorem changed summation constrains

I know that a factorized polynomial can be written in the following way: $$(x_{11}+...+x_{1n})^{a_1}...(x_{n1}+...+x_{nn})^{a_n}=\sum_{k_{11}+...+k_{1n}=a_1\\\quad\quad\vdots\\k_{n1}+...+k_{nn}=a_n\\}\...
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2 votes
2 answers
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Simplify a sum with a product and multinomial coefficient

This is a follow-up question to my previous post, where I've got a great help! (I created a new one to avoid editing the original one). Can the following sum be simplified? $$ \sum_{\substack{k_1 + ...
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Confusion regarding the proof of the Multinomial theorem

We saw the following theorem in class: If $n \in \mathbb{N}$ and $z_1, \dots, z_m \in \mathbb{C}$ we have: $ (z_1 + \dots + z_m)^n= \sum_{k_1+\dots+k_m=n} \binom{n}{k_1, \dots, k_m} z_1^{k_1} \dots ...
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3 votes
1 answer
202 views

Simplifying the sum of a product of multinomial coefficients

From the multinomial theorem the following holds $$ \sum_{k_1 + k_2 + \ldots + k_m=n} {n \choose k_1, k_2, \ldots, k_m} = m^n $$ I have the following sum $$ \sum_{\substack{k_1 + k_2 + \ldots + k_m ...
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Find number of terms and coefficient of $x^5$ in $(1+x+x^2)^7$

I tried using the multinomial theorem to find general term ie. $$T_n =\frac{7!}{a!b!c!} 1^a \cdot x^b \cdot x^{2c} $$ Now $b+2c=5$ and $a+b+c=7$ The second equation can be interpreted as distributing $...
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Let $(2x^2 +3x+4)^{10} = \sum _{r=0}^{20} a_rx^r$. Then find $\frac{a_7}{a_{13}}$

I tried using the multinomial theorem where the n’th term is $$\frac{10!}{r_1!r_2!r_3!}\times 2^{r_1}3^{r_2} 4^{r_3} x^{2r_1 + r_2}$$ ($r$ is an integer) Where $$r_1+r_2+r_3=10$$ and For first part ie....
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problem with Multinomial Theorem [closed]

I am trying to find the constant term in this expansion: $$\bigg(1 + x + 2y^{2} - \frac{1}{x^{2}y}\bigg)^{15}$$ I have been trying for hours but I hit a wall when working out the values $(r_1, r_2, ...
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1 vote
1 answer
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The $q$ Multilinear Theorem

Let $R$ be the skew polynomial ring $k_\mathfrak{q}[x_1,\ldots,x_m]$ where $x_ix_j=qx_jx_i$ with $q\in k^*$ and for all $i<j$. The $q$ Multinomial Theorem states that $$(x_1+\ldots+x_m)^r=\Sigma_{...
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2 votes
2 answers
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Can the following expression be related to the multinomial formula?

The following formula $$(n,p,q,r\ \text{odd})\quad \sum_{\genfrac{}{}{0pt}{1}{p\leq q \leq r}{p+q+r =n}} \frac{n!}{p!\, q!\, r!} \times \begin{cases} 1 & \text{if}\ p< q <r\\ \frac{1}{2} &...
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How many rational elements are in the multinomial? $(\sqrt[3]{x}+\sqrt[5]{x}+2\sqrt{x})^5$

How many rational elements are in the multinomial? $(\sqrt[3]{x}+\sqrt[5]{x}+2\sqrt{x})^5$ According to the multinomial theorem, $\sum_{i+j+k=5}^{}\frac{5!}{i!j!k!}2^k\ x^{\frac{1}{3}i+\frac{1}{5}j+\...
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1 vote
0 answers
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Why is $P(X=x, Y=y, Z=z)$ = $\frac{n!}{x!y!z!}$?

So I was reading this question: What are the chances that out of $n$ people playing rock, paper, scissors, only two choices are picked? and saw this type of answer in other place as well. The: But ...
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2 votes
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Prove that ${a_{1} +2a_{2} +3a_{3}+\cdots+np\ a_{n}}_{p} \ =\frac{np}{2}( 1+p)^{n}$

In expansion of ${\ \left( 1+x+x^{2} +\cdots+x^{p}\right)^{n} =a_{0} +a_{1} x+a_{2} x^{2} +\cdots+a_{np}} x^{np}$, prove that ${a_{1} +2a_{2} +3a_{3} +\cdots+np\ a_{n}}_{p} \ =\frac{np}{2}( 1+p)^{n}$. ...
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What is the asymptotic growth of the terms in this expression?

Let's say we have $k$ variables in this expression: $(n_1+n_2+n_3+...+n_k)^2$. When you expand it, you get something like $n_1^2+n_1n_2+n_2^2+...$ , using the multinomial theorem. Now, if we multiply ...
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Application of multinominal theorem for a tricky sum

In our lecture notes, it says that one can compute the sum $$\sum_{\substack{k_1,k_2,\dots,k_M\geq0:\\k_1+\dots+k_M=N}} \binom{N}{k_1,\dots,k_M} \left(\binom{k_1}{2} + \dots + \binom{k_M}{2} \right)^2$...
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  • 358
3 votes
2 answers
140 views

Proof of Sum of Trinomial Coeffs = $3^n$

I'm trying to prove that the sum of trinomial coeffs $\sum_{l+k+m=n}^n \binom{n}{l,k,m} = 3^n$. I tried by induction and got the step as follows: $$ \begin{split} \sum_{l+k+m=n+1}^{n+1} \binom{n+1}{l,...
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Multinomial type finite sum

In a problem related to the study of the Weil-Petersson volume of the moduli space of bordered Riemann surfaces of genus $g$ with $m$ geodesic boundaries, all of length $\ell > 0$, I've encountered ...
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What is the size of a multinomial?

The answer to this question uses the phrase "multinomial of size". What is the definition of the size of a multinomial? They are using a negative multinomial.
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PGF of negative multinomial expansion

I have found the formula for the Probability Generating Function of negative multinomial distribution in Definition 8.1 of this chapter (https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118445112....
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1 vote
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Negative multinomial expansion

I would like to know the standard form of the negative multinomial expansion i.e. $(x_1 + x_2 + \ldots + x_p)^{-n}$. I understand that I can probably derive something by applying the negative binomial ...
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1 vote
2 answers
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Why do we divide by the count of each letter in questions of the form "how many words can I form from the letters in"?

For example, how many words can we form from the letters in the word google? First I thought you counted how many different letters there are in this case 4, therefore in each spot (6 spots) there are ...
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1 answer
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$a_{m, n}$ is coefficient of $x^n$ in expansion $(1+x+x^2)^m.$ Prove $0\leq \sum_{i = 0}^{\left\lfloor 2k/3\right\rfloor} (-1)^ia_{k-i,i} \leq 1$

Problem Statement: Let $a_{m, n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^m.$ Prove that for all $k\geq 0$, $$0\leq \sum_{i = 0}^{\left\lfloor 2k/3\right\rfloor} (-1)^ia_{k-i,i} \...
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How to find the coefficient of $x^{l}$ in the expansion of $(1+x+x^{2}+...+x^{n})^{m}$ for some given $l$? [duplicate]

Suppose I'd like to find the coefficient of $x^{l}$ in the expansion of $(1+x+x^{2}+...+x^{n})^{m}$, where $n$ and $m$ are given positive integers, for some given integer $l$ such that $n < l < ...
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Problem related to multinomial expansion.

How do I expand $(a_1+a_2+a_3+.....+a_k)^3$ where $a_i \in \Bbb R $ for $i = 1,2,...,k $?
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1 vote
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finding the coefficient of ${t}^{20}$ in the expansion of ${({t}^{3}-3{t}^{2}+7t+1)}^{11}$

I saw a question in my textbook.I tried to solve it using multinomial theorem.However,i stuck in somewhere. The question is: find the coefficient of ${t}^{20}$ in the expansion of ${({t}^{3}-3{t}^{2}+...
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2 votes
2 answers
177 views

Find the coefficient of ${x}^{20 }$ in ${({x}^{2}+{x}^{3}+{x}^{4}+{x}^{5}+{x}^{6})}^{5}$

I saw a question in my textbook, the solution of this question exists in my textbook. However , its solution is very long.I tried to solve it in different way but i do not know whether it is true or ...
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0 votes
1 answer
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Methods to find coefficient of a term in sum of polynomials

I have a polynomial sequence that I would like to sum, that goes as follows: $1 - (y-1) + (y-1)^2 - (y-1)^3 + ... + (y - 1)^{17}$ which is basically $\sum^{17}_{r = 0} (-1)^r (y-1)^r$ I would like to ...
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1 vote
1 answer
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Understanding a variant on the multinomial theorem in a commutative ring with unity

This post concerns Chapter 1 section "The Multinomial Theorem" on pages 65-67 of Analysis I by Amann and Escher. Excerpts from text: The part that I can't understand is the equation with the ...
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1 vote
1 answer
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Multinomial - Sum of coeffecients with even powers

Let P be a polynomial given by $P(x_1,x_2,x_3, \ldots,x_n) = (k+x_1+x_2+\ldots +x_n)^m$. Find the sum of all coefficients of the terms of the polynomial which have even powers in each of the $n$ ...
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1 vote
1 answer
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Prove $\frac{n !}{(n-k) !} \cdot k^{n-k}$

$$\sum_{n_{1}+n_{2}+\cdots+n_{k}=n}\binom{n}{n_{1}, n_{2}, \dotsc,n_{k}} \cdot n_{1} n_{2} \cdots n_{k}=\frac{n !}{(n-k) !} \cdot k^{n-k}$$ I try this but I don't know if I am right: \begin{align} &...
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1 vote
0 answers
46 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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2 votes
1 answer
547 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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3 votes
2 answers
72 views

Prove combinatorics equality?

Assume $j$ is fixed, prove the following: $$\sum_{i}\binom{n}{i, j, n-i-j} = 2^{n-j}\binom{n}{j}$$ So the left hand side reminds me the multinomial theorem and we can think of a long sequence word ...
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  • 287
4 votes
2 answers
202 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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0 answers
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Multi Index Power Series

What is the closed-form expression for the summation $$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$, $m$ and $p$?...
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-1 votes
1 answer
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Find coefficient of $x^{10}$ in $\left(1−x^7\right) \left(1−x^8\right) \left(1−x^9\right) (1−x)^{-3}$ [closed]

find the coefficient of $x^{10}$ in this expansion: $$ \left(1−x^7\right) \left(1−x^8\right) \left(1−x^9\right) (1−x)^{-3} $$ Please help me solve this question
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