# Questions tagged [multinomial-coefficients]

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

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### Solution Explanation for 2016 AMC 10A Problem 20

This is a combinatorics problem from the 2016 AMC 10A [ Problem 20 ] For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains ...
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### Series about coefficients of multiplicative inverse of power series

Let be an integer $d\geqslant 2$ and a real number $L\in(0,1)$. I consider the following formal power series $$T(x) := 1-L\,\sum_{1\leqslant j < d} x^j =: \sum_{n\geqslant 0} {a_n}x^n$$ with $a_0=1$...
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### Extracting coefficients with the power series $(1-x)^{-n}$

Given polynomials of the form $$(1+x+x^2+x^3+\cdots+x^k)^n$$ We can calcualte the coefficients by writing it in the form $$(1-x^{k+1})^n \over (1-x)^n$$ and using the power series $(1-x)^{-n}$, as ...
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### Combinatorial interpretation of the multinomial coefficient as a product of binomial coefficients

\begin{align}& \binom{n}{k_1,k_2,\dots,k_m}\\&=\frac{n!}{k_1!k_2!\cdots k_m!}\\&=\binom{k_1}{k_1} \binom{k_1+k_2}{k_2}\cdots \binom {k_1+k_2+\cdots+k_{m}}{k_{m}}\end{align} Is there ...
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### Prove $\sum_k{{a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}(-1)^k}=\dfrac{(a+b+c)!}{a!b!c!}$ [duplicate]

How can I prove this: $$\sum_k{{a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}(-1)^k}=\dfrac{(a+b+c)!}{a!b!c!}$$ I know I should avoid no clue questions, but really I have no idea about this one. ...
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### Question: Concerning Simplifying Random Walk from 2D to 1D

I have a question that has been confusing me. For a 1D random walk in the x-direction I was told that the multinomial coefficient is given by: $$C(N,k_x) = \frac{N!}{k_x!(N-k_x)!} \tag{1}$$ In Eq. 1, ...
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### Weighted sum of specific multinomial coefficients

Let $A$ and $b$ be nonnegative integers and consider the sums $$\sum\limits_{c=0}^{b/2}\frac{1}{4^c}\binom{A}{c,b-2c,A-b+c}$$ and $$\sum\limits_{c=0}^{b/2}\frac{c}{4^c}\binom{A}{c,b-2c,A-b+c}.$$ I ...
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### Calculating Coefficents of a single variable polynomial [duplicate]

Given: $$(1+x+x^2+x^3+\cdots+x^k)^n$$ Is there a formula to calculate the coefficient of $x^a$ (where $a$ can be any integer value less than $k^n$) that's more efficient than grinding through ...
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### Coefficient of $x^k$ in polynomial

Let $k, n, m \in \mathbb{N}, k \le n.$ Find the formula for coefficient of $x^k$ in $(x^n + x^{(n-1)} + ... + x^2 + x + 1)^m$. answer is in this question: faster-way-to-find-coefficient-of-xn-in-1-x-...
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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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### 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 ...
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### But why is this a legitimate way of writing out trinomial expansion?

So recently, I asked this question asking if $\sum_{r=0}^n\sum_{s=0}^r\dfrac{n!a^{n-r}b^{r-s}c^s}{s!(n-r)!(r-s)!}$ was a legitimate way of expanding $(a+b+c)^n$ Multi-binomial theorem When working in ...
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### Sum of multinomial coefficient

I want to show that $$\sum_{j,k,j+k\leq n} 3^{-n}\frac{n!}{j!k!(n-k-j)!}=1.$$ I tried to do it by induction, but this clearly seems the wrong approach because the denominator becomes very hard to deal ...
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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 ${n+m-1 \choose m-1}$ (...
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### Finding number of integer solutions by generating functions. [closed]

So, we got the worst professor ever who didn't tell us actually how to solve generating functions to obtain coefficients but just ran over some examples giving random theorems and results to obtain ...
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### How to reduce overcounting in group assignment problem when objects are not distinct [duplicate]

Section 4 in https://web.stanford.edu/class/archive/cs/cs109/cs109.1206/lectureNotes/LN02_combinatorics.pdf has an explanation of multinomial coefficient to address ...
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### Is there a parametric distribution family for natural-numbered random vectors $X=(X_1, X_2,...,X_N)$ with a simplex-like constraint
Given $a\in\mathbb{N}^N, c\in\mathbb{N}$, is there a way to generate random vectors $X=(X_1, X_2,...,X_N), X_n\in\mathbb{N}$ with constraint that $a\cdot X=c$ (inner product)? If so, how will be these ...