# 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 ...
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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, $0 \leq x_3\leq n$ $\Rightarrow n \leq x_1 + x_2 \leq 2n$ ...
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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 ...
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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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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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