# Questions tagged [multinomial-coefficients]

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

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### In how many ways can I split a group of 15 children into 5 groups of size $1,2,3,4,5$ [closed]

In how many ways can I split a group of 15 children into 5 groups of size $1,2,3,4,5$? I thought about $^{15}C_1+ ^{14}C_2+....+ ^5C_5$ , but it wont be consistent , someone can help please?
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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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### Multiplicity of integer partitions in iterative process

Let $(M_k)_{k\geq0}$ be a sequence of multisets. The multiset $M_0=\{[\:]\}$ has only one element, which is an empty sequence. For positive $k$, $M_k$ is a multiset of sequences of integers sorted in ...
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### Number of ways of splitting set into disjoint subsets

How many distinct ways are there of splitting a finite set $X$ with $|X| = n$ into $k$ disjoint subsets of sizes $n_1, n_2, \dots, n_k$ with $\sum_{i=1}^k n_i = n$? I would think the answer should be: ...
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### How many ways are there to distribute eight different toys among four children if the first child gets at least two toys?

My work: $e_1 + e_2 + e_3 + e_4 = 8$ let $e_1\geq2$ with no constraint on the other $e_i$'s we want to find coefficient of $\cfrac{x^8}{8!}$ with this I've found the exponential generating function to ...
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### Find coefficient of $x^r$ in $(x^5+x^6+x^7+…)^8$

My work: We can rewrite the generating function $(x^5+x^6+x^7+...)^8$ as $x^{40}(1+x+x^2+...)^8$ We are looking for $x^{r-40}$ coefficient in the new generating function $(1+x+x^2+...)^8$ We can ...
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### Find the coefficient of $x^{12}$ in $(1+x)^{-1}$

My work: $(1 + x)^{-1} = \left(\cfrac{1-x^2}{1-x}\right)^{-1} = \left(\cfrac{1-x}{1-x^2}\right)^{1} =(1-x)^1(1-x^2)^{-1}$ So we do $C(1,0) \cdot C(-1,6) = 0$ to find all the ways to get $x^{12}$ But ...
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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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### How many positive integral solutions exist for $2a+3b-c = 0$ where $a$ ranges from $0$ to $5$, $b$ from $0$ to $10$ and $c$ from $0$ to $40$?

I was stuck with this particular problem. I tried finding a solution by attempting to find the coefficient of $x^0$ in $(1+x^2 +\dots+x^{10})(1+x^3 +\dots+x^{30})(1+x^{-1} +\dots+x^{-40})$ but for ...
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### Bound on function of sum of powers

Let $(x_1, \ldots, x_k)\in R^k$ and $n=n_1+\ldots +n_k$, with $n\in N_0$ and $0\leq n_i\leq n$ Consider function $M_n=\sum_{i=1, \textit{number of terms is$2\ell+1$}}^kx_i^{n_i}$-function with odd ...
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### Combination with at least n elements

I have the following problem: There is a set of $30$ elements. It must be split in $3$ subsets. However, each subset must have, at least, $9$ elements. How can I count how many ways are there to ...
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### I need a combinatorial proof of $\sum_{n_1+n_2+n_3=n} \binom{n}{n_1, n_2,n_3}(-1)^{n_2} = 1$

$$\sum_{n_1+n_2+n_3=n} \binom{n}{n_1, n_2,n_3}(-1)^{n_2} = 1$$ I tried labeling $n$ objects 1 or 2 or 3 and subtracting even numbers of 2 from odd numbers of 2, but couldn't go further. Is there a ...
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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....
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 ...