# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $p$?...
### 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