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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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Finding number of dissimilar terms in an expansion

The question is The number of dissimilar terms in the expansion of $(1+x^3+x^4)^4$ is ? Upon using the formula $^{n+r-1}C_{r-1}$ or using permutation & combination, I am getting 15 as the answer ...
ADITYA DAS's user avatar
1 vote
2 answers
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Number of integral solutions for $x_1 + x_2 - x_3 = n$ where $n \geq x_1 , x_2 , x_3 \geq 0$

I have been asked Integral solutions for $x_1 + x_2 - x_3 = n$ where $n \geq x_1 , x_2 , x_3 \geq 0$. My approach: We have, $0 \leq x_3\leq n$ $\Rightarrow n \leq x_1 + x_2 \leq 2n$ ...
QuantumQuipster's user avatar
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Number of monotonically increasing functions such that $f(i)\le i$.

Problem: Consider $n \in \mathbb{N}^+$, set $A = \mathbb{N}^+ _{\leq n}$. Find the number of monotonically increasing functions $f: A → A $ such that $f(i) \leq i$. I tried using the multinomial ...
Trulaug's user avatar
3 votes
1 answer
82 views

Showing the quintessential logarithm property using the Maclaurin series of $\log$

For $-1\le x<1$, we have $$\log(1-x) = -\sum_{k=1}^{\infty} \frac{x^{k}}{k}\\$$ Taking $a,b$ with $|a|,|b|<1$ and $(1-a)(1-b)\le2$, on the function side clearly we have $$\log(1-a)+\log(1-b) = \...
Integrand's user avatar
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4 votes
1 answer
340 views

Why do Bell Polynomial coefficients show up here?

The multinomial theorem allows us to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + {x_4} + ...} \right)^n}$. I am interested in the coefficients when expanding ${\left( {\sum\...
Bear's user avatar
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2 answers
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If $(1+px+x^2)^n=1+a_{1}x+a_{2} x^2+....+a_{2n}x^{2n}$, then prove that $(np-pr)a_{r}=(r+1)a_{r+1}+(r-1-2n)a_{r-1}$ for $1<r<2n$

If $(1+px+x^2)^n=1+a_{1}x+a_{2} x^2+....+a_{2n}x^{2n}$, then prove that $(np-pr)a_{r}=(r+1)a_{r+1}+(r-1-2n)a_{r-1}$ for $1<r<2n$ My try: I tried putting $r=2$ and solved the problem and verfied ...
mathophile's user avatar
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$(1+2x+3x^2)^{50}=a_{0}+ a_{1}x+ ... +a_{100}x^{100}$ Find ratio between $a_{51}$ and $a_{49}$.

$$(1+2x+3x^2)^{50}=a_{0}+ a_{1}x +...+ a_{100}x^{100}$$ Find ratio between $a_{51}$ and $a_{49}$. Another sub question asked is to find the relation between $a_{n}, a_{n-1}, a_{n-2}$ My approach I ...
Patrick Schick's user avatar
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Multinomial Theorem expansion in Combinations Problem

While studying application of Multinomial theorem in PnC I got stuck in two questions : In how many ways the sum of upper faces of four distinct dice can be six ? The textbook gave the following ...
Mokshit Arora's user avatar
2 votes
1 answer
146 views

Proving the Number Theory Property: $\delta(n^k) = \delta(\delta(n)^k)$ for natural $n$ and $k$

I'm working on a number theory problem for the Regional Mathematical Olympiad, Stage 2 of the Indian Olympiad Programme, and it's been challenging to solve. The problem is as follows. A function $\...
Yatharth Shrivastava's user avatar
2 votes
1 answer
116 views

Central Binomial Coefficients and Multinomial Coefficients

Premise I was looking at the multinomial coefficients when selecting by a specific rule. Then analyzing the sum. Given the multinonial theorem ($n > 0$): $$ (x_1+\ldots+x_n)^n = \sum_{k_1+\ldots+...
tkellehe's user avatar
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Multinomial theorem for a power series

I was wondering if there is a version of the multinomial theorem for the expression: $$ (1+\sum_{k=1}^\infty a_k x^k)^n. $$ Thanks in advance.
Ludwig's user avatar
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3 answers
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The proof by induction of the multinomial theorem

I looked at the proof by induction of the multinomial theorem on Wikipedia and do not understand how to get the last step. Specifically, I do not know why this equality is true: $$\sum_{k_1 + k_2 + \...
user1181399's user avatar
1 vote
3 answers
344 views

Five different games are to be distributed among $4$ children randomly. The probability that each child get at least one game is?

Here's a question :- Five different games are to be distributed among 4 children randomly. The probability that each child get at least one game is? To find the answer, we will have to find the sample ...
Avish Bhatia's user avatar
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0 answers
137 views

In the multinomial expansion of $(a+b+c+d)^8$, how many terms (monomials) have coefficient $\begin{pmatrix} 8 \\ 2,4,0,2\end{pmatrix}$?

Can someone please explain why the ansewr is 12? My current working: Using the multinomial theorem, each term in the expanded (unsimplified) form would be uniquely determined by its distinct ordered ...
Jason Xu's user avatar
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63 views

When not to use $\binom{n + r − 1} {r− 1}$?

The number of non-negative integral solutions of the equation $x_1+x_2+x_3+....+x_r = n$ is $\binom{n + r − 1} {r− 1}$ I tried using it in the following two questions. Let $n_1<n_2<n_3<n_4&...
XZCY's user avatar
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2 votes
4 answers
262 views

A multinomial theorem for polynomials of one variable

Theorem (Multinomial Theorem): For any positive integer $m$ and any non-negative integer $n$, the following formula holds: $$ \left(\sum_{i=1}^m x_n\right)^n = \sum_{\begin{gathered} k_1+k_2+\dots+k_m=...
Rusurano's user avatar
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doubt in the chapter of multinomial theorem

the question stated to find the sum of all coefficients divisible by 37 in the expansion of $ (2x+y+z)^{37} $ I proceeded by writing out $ (2x+y+z)^{37} $ = $ \sum_{0 \le p,q,r \le 37} $ (${ (37!) ...
Devang Tripathi's user avatar
1 vote
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Total number of vertices on a pascal m-plex [duplicate]

Let m > 0 be a number of terms of a polynomial and n ≥ 0 be a power the polynomial is raised to. The number of terms in the multinomial $(x_1 + x_2 + ... x_m)^n$ is given by ${n+m-1 \choose m-1}$ (...
Bobuji's user avatar
  • 11
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1 answer
168 views

Calculating sum of coefficients of all terms of multinomial expansion

Working through some combinatorics problems and am currently working on one involving a trinomial expansion of $$(x+y+z)^6$$ The question asks: How many terms are in this expansion? What is the ...
Numerical Disintegration's user avatar
0 votes
1 answer
43 views

Simplify 4-element multinomial distribution where there are only 2 unique probabilities

I am having some trouble understanding if the following expression can be simplified \begin{equation} \binom{m}{b, c'-b, c-b, m+b-(c'+c)} P_s^{m+2b-(c'+c)} (1-P_s)^{((c'+c)-2b)} \end{equation} ...
jamessud's user avatar
1 vote
1 answer
205 views

Relation between series expansion coefficients and original function

So I have a function of the form: $$f(x) = \frac{1}{1-x p_{2m}(x)}$$ where $p_{2m}(x)$ is a polynomial of degree $2m$. I can expand the function around $x=0$ as: $$f(x) =\sum_{n=0}^{\infty}c_n x^n$$ ...
Fra's user avatar
  • 208
0 votes
1 answer
72 views

answer verification, permutations (ways to make n words out of m letters)

How many ways are there to make different words (not necessarily meaningful) by changing order of letters in the word $b_1b_1...b_1b_2b_2...b_2...b_mb_m...b_m$, where $b_1$ appears $k_1$ times, $b_2$ ...
John Doe's user avatar
  • 502
2 votes
0 answers
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Multinomial theorem and differential of a function.

Let $\alpha=(\alpha_1,\dots \alpha_n)$ be a multi-index of order $\lvert \alpha \rvert=\alpha_1+\cdots+\alpha_n $ and let $u\colon\mathbb{R}^n\to \mathbb{R}$ be a sufficiently regular function. I must ...
NatMath's user avatar
  • 162
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1 answer
377 views

How many collected terms are in the expansion of $(x+y+z)^{10} (w+x+y+z)^2$?

How many collected terms are in the expansion of $(x+y+z)^{10} (w+x+y+z)^2$? Hi, I'm trying to solve this problem as study material for discrete mathematics and I'm not quite sure how. I got 235 terms ...
Gift G.'s user avatar
2 votes
2 answers
177 views

Multinomial identity

Consider that $p_0,\dots,p_m$ are probabilities such that $\sum\limits_{i=0}^{m}p_i=1$. I would like to prove that \begin{align} \textstyle n\sum\limits_{i=0}^{m}ip_i=\sum\limits_{k_0+\dots +k_m=n;\...
Waney's user avatar
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3 votes
1 answer
184 views

Alternate multinomial theorem for $\frac{d^n}{dx^n}\prod\limits_{k=1}^m f_k(x)$ without $\sum\limits_{k_1+\dots+k_m=n}$ nor Kronecker delta.

The generalized product rule complicates putting series coefficients into closed or hypergeometric form. There are 2 forms with Lagrange $n$th derivative notation and the multinomial $\binom n{n_1,\...
Тyma Gaidash's user avatar
1 vote
0 answers
46 views

I need help extending the Multinomial Theorem to Polynomials

I have been pondering the question, what does $f(x)$ look like if $f(x)=\left(\frac{d^{w}}{dx^{w}}f(x)\right)^n$ and $f(x)$ is a polynomial. If you're given w, then I believe that there are a limited ...
John Whitacre's user avatar
2 votes
3 answers
434 views

Calculating Coefficients of an N Degree Polynomial raised to an Arbitrary Power

Suppose you have $(a_0+a_1x+a_2x^2+...+a_nx^n)^k$, and you want to expand and find a formula for the coefficients $\beta_j$ such that $\beta_j$ is the coefficient of the $x^j$ term. I understand that ...
mbohde2015's user avatar
1 vote
1 answer
158 views

I have the sequence: $1, 4, 10, 16, 19, 16, 10, 4, 1$. What is the formula to get the $i^{th}$ term of the sequence for $i=1,2,\dots,9$

I've been trying to derive this formula for quite some time now with little progress. I've seen concepts such as Pascal's pyramids and Pascal's simplices mentioned throughout my research; however, I ...
rizevenk11's user avatar
0 votes
2 answers
207 views

Constrained multinomial theorem removing terms from sum

The multinomial theorem dictates that $$\sum_{\mu_0+\mu_1+\cdots+\mu_M=N}\binom{N}{\mu_0,\mu_1,\cdots,\mu_M}x_0^{\mu_0}x_1^{\mu_1}\cdots x_M^{\mu_M}=(x_0+x_1+\cdots+x_M)^N.$$ Here, the multinomial ...
Quantum Mechanic's user avatar
2 votes
3 answers
85 views

Coefficient problem using multinomial theorem

i want to solve this: consider $(x+y+z)^n$, let $n=1000$ the coefficient of $x^{320}y^{410}z^{270}$ can be written as $\binom{a}{b} \cdot \binom{c}{d}$. find $a,b,c,d \in \mathbb{N}$ my attempt is ...
Fermatto's user avatar
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4 votes
0 answers
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Multinomial theorem with multivariate terms?

Let $S=\{a,b,c,d,...\}$. Let $P_n=(abc+abd+acd+...+ab+ac+ad+...a+b+c+d...)^n$. In addition, there's the condition that for all variables, $x^n=x$ (maybe it'll be easier without this?). Is there ...
DrownedSuccess's user avatar
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0 answers
145 views

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}(...
AtKin's user avatar
  • 608
2 votes
1 answer
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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>...
Leonardo's user avatar
6 votes
5 answers
235 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 ...
Jack Moresy's user avatar
2 votes
1 answer
122 views

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 ...
user110391's user avatar
  • 1,129
3 votes
2 answers
100 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 = ...
Flyrom's user avatar
  • 73
6 votes
1 answer
266 views

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 $\left(1 + x^{2} + x^{4} + x^{5}\...
V.G's user avatar
  • 4,196
0 votes
2 answers
134 views

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 ...
Haoran Chen's user avatar
-1 votes
2 answers
119 views

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 )^...
I Am A Bad Programmer's user avatar
0 votes
2 answers
62 views

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 ...
Gaurav Kumar's user avatar
7 votes
2 answers
167 views

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 \...
a_guest's user avatar
  • 211
1 vote
1 answer
118 views

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,...
s1624210's user avatar
3 votes
1 answer
85 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 \...
Abhishek A Udupa's user avatar
1 vote
1 answer
48 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 ...
arow257's user avatar
  • 334
3 votes
0 answers
75 views

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\\}\...
meneken17's user avatar
  • 162
2 votes
2 answers
165 views

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 + ...
SugerBoy's user avatar
  • 703
0 votes
1 answer
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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 ...
user avatar
4 votes
1 answer
509 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 ...
SugerBoy's user avatar
  • 703
0 votes
2 answers
186 views

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 $...
Aditya 's user avatar
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