Questions tagged [multinomial-coefficients]

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

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69 views

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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40 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 )^...
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2answers
107 views

Question from pathfinder for Olympiad mathematics 2 [duplicate]

If $p$, $q$, $r$ are the real roots of equation $x^3-6x^2+3x+1=0$, determine the possible value of $p^2q+q^2r+r^2p$. My Attempt: $p+q+r=6 (1)$ $pq+qr+pr=3 (2)$ $pqr=-1 (3)$ Multiplying (1) ...
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0answers
95 views

Is this relation already discovered?

$$ \sum_{d \mid (n,k_1,k_2, \dots,k_m)}\mu(d)\binom{n/d}{k_1/d, k_2/d, \dots, k_m/d} \equiv 0 \pmod n $$ where $\mu$ is the Moebius mu function. I've found above interesting divisibility properties. I'...
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2answers
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Probability that each of the players gets an ace

I was reading Introduction to Probability, 2nd Edition, and the following question appears as exercise $60$ in the first chapter: A well-shuffled 52-card deck is dealt to $4$ players. Find the ...
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19 views

Probability of sampling x elements of a given type in n attempts from a bin with s initial elements of that type out of total k elements.

Which is the probability of sampling exactly $x$ elements in $n$ attempts of a certain type from a bin knowing that the bin had originally $s$ elements of that type out of total $k$ elements. with ...
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1answer
57 views

How many terms are there containing the term $xyk^2$ in the expansion of $(2x-y+t+3z+4k)^8$

How many terms are there containing the term $xyk^2$ in the expansion of $(2x-y+t+3z+4k)^8$ such as $xyk^2t^2z^2$ or $xyk^2t^4z^0$ or $xyk^2z^3t$ etc. I made up this question and calculated it , but ...
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1answer
25 views

Notation for set containing combinations of sets given by multinomial coefficient number of options.

Assume there are sets $A_1,A_2,\dots,A_n$. Let $m\leq n$ and now partition $\{1,\dots,n\}$ into $m$ subsets $N_1,\dots,N_m$. So, there are basically $$\sum_{l_1+\dots+l_m = n} {{n}\choose{l_1,\dots,...
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3answers
80 views

Number of derangement on $(1, 1, 2, 2, 3, 3, 4, 4)$

How many 8-tuples of $[4]^8=\{1,2,3,4\}^8$ are there s.t. every number in $[4]$ appears exactly twice, and $i$ never appears on the $i$th place for all $i\in[4]$? There are $8!/16$ different tuples ...
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2answers
104 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 \...
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1answer
59 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 \...
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0answers
34 views

proving the multinomial theorem $\frac{\left(\frac{n!}{n_r!}\right)}{n_z!}$

I'm working through an introduction to probability book and although it doesn't ask to proof the theorem in the title, in fact, that is how I imagine it to look so it may be wrong. What it asks to ...
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0answers
49 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\\}\...
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2answers
90 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 + ...
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1answer
124 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 ...
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2answers
68 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 $...
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1answer
76 views

Simplify sum $\sum_{m_1,m_2,\ldots,m_k}{p \choose m_1\; m_2 \; \ldots \; m_k}\Pi_{i=1}^k (2m_i-1)!!$ for $p\to \infty$

Let $1 \le k \le p$ be integers. Define $$ S_{p,k} := \sum_{m_1,m_2,\ldots,m_k}{p \choose m_1\; m_2 \; \ldots \; m_k}\Pi_{i=1}^k (2m_i-1)!!, $$ where the sum if over all nonnegative $k$-partitions $(...
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2answers
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Question: For what value of $c$ will $P(x)= -2x^3+cx^2-5x+2$ have the same remainder when devided by $x+1$ or $x-2$?

For what value of $c$ will $P(x)= -2x^3+cx^2-5x+2$ have the same remainder when devided by $x+1$ or $x-2$? No idea where to start with this one. Can't use the remainder theorem to find the remainder ...
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1answer
28 views

For what values of n, will $1 + 2^n + 3^n + 4^n$ be divisible by $5$ where $0 \leq n \leq 100$

There's this problem that I am trying to find the solution for: For what values of n, will $1 + 2^n + 3^n + 4^n$ be divisible by $5$ where $0 \leq n \leq 100$ So that doesn't hold for $n=0$, but ...
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4answers
141 views

Find the coefficient of $x^{16}$ in $(1 + x + x^2 + x^3 + x^4 + x^5)^4$

We are supposed to find the coefficients of it, I wanted to know if my approach is right here. The final answers seems a bit iffy. $$(1+x+\dots+x^5)^4=\left(\frac{1-x^6}{1-x}\right)^4=(1-x^6)^4(1-x)^{-...
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1answer
35 views

The $q$ Multilinear Theorem

Let $R$ be the skew polynomial ring $k_\mathfrak{q}[x_1,\ldots,x_m]$ where $x_ix_j=qx_jx_i$ with $q\in k^*$ and for all $i<j$. The $q$ Multinomial Theorem states that $$(x_1+\ldots+x_m)^r=\Sigma_{...
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2answers
67 views

Can the following expression be related to the multinomial formula?

The following formula $$(n,p,q,r\ \text{odd})\quad \sum_{\genfrac{}{}{0pt}{1}{p\leq q \leq r}{p+q+r =n}} \frac{n!}{p!\, q!\, r!} \times \begin{cases} 1 & \text{if}\ p< q <r\\ \frac{1}{2} &...
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1answer
40 views

Number of ways to arrange three 1s, two 2s, two 3s and one 4?

Number of ways to arrange three 1s, two 2s, two 3s and one 4? The order doesn't matter. The direct answer from the book is $\frac{8!}{3!2!2!1!}$, but is there another way to do it?
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3answers
84 views

How many words can be created by using the letters of “MATHEMATIK” when no same letters can be next to each other?

The word MATHEMATIK (in german) has 10 letters. I want to know how many different words can be created by using the letters of this word. However, the same letter can never be next to each other. So ...
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1answer
47 views

Maximizing multinomial distribution with constraints using Lagrange multipliers

Use methods of Lagrange multipliers to maximize $$W (N_{1}, \dots, N_{M}) = \frac{N!}{\prod_{j=1}^{M} N_{j}!}$$ under the constraints $\sum N_{j} = N =$ constant, $\sum E_{j} N_{j} = \mathcal{E} =$ ...
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2answers
27 views

how to build a formula for calculation / coefficient(s) for set of descending numbers

I have been years away from my math class. Now I would like to figure out the following. I am writing a simple javascript program where some tabular data is output to PDF file. The program will ...
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2answers
113 views

Deriving Generating function for centered trinomial coefficients

Let $c_n$ denote the $n$-th center trinomial coefficient (OEIS sequence here). It appears they cannot be generated by a linear recurrence relation, so how should I go about to find the generating ...
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0answers
47 views

What is the meaning of the number $\frac{k!}{a_1!…a_n!}$?

It says that this is the number of $k$ tuples $(i_1,...,i_k), 1\leq i_k \leq n$ where the index $v$ appears exactly $a_v$ times $(v=1,..,n, a_1+...+a_n=k)$ Can someone give me an explaination how this ...
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0answers
21 views

Nasty alternating sum with multivariate beta function.

Given $\mathbf x$ in the d-dimensional simplex (i.e. $x_i > 0$ for all $i \in \{1,...,d\}$ and $\lvert \mathbf x \rvert \le 1$), $\alpha \in \mathbb{R}_+$, and a given multiindex $\mathbf p \in \...
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1answer
69 views

Prove that ${a_{1} +2a_{2} +3a_{3}+\cdots+np\ a_{n}}_{p} \ =\frac{np}{2}( 1+p)^{n}$

In expansion of ${\ \left( 1+x+x^{2} +\cdots+x^{p}\right)^{n} =a_{0} +a_{1} x+a_{2} x^{2} +\cdots+a_{np}} x^{np}$, prove that ${a_{1} +2a_{2} +3a_{3} +\cdots+np\ a_{n}}_{p} \ =\frac{np}{2}( 1+p)^{n}$. ...
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1answer
27 views

Statistics of a random process derived from multinomial distribution

Suppose we have a random process defined on set with $S=\{1, 2, \ldots, N\}$, where at each time step $1 \leq t \leq M$, you select a random variable $ X_t \in S$ with equal probability. I am ...
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2answers
86 views

Proof of Sum of Trinomial Coeffs = $3^n$

I'm trying to prove that the sum of trinomial coeffs $\sum_{l+k+m=n}^n \binom{n}{l,k,m} = 3^n$. I tried by induction and got the step as follows: $$ \begin{split} \sum_{l+k+m=n+1}^{n+1} \binom{n+1}{l,...
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1answer
93 views

What is the number of partitions of a set of size $16$ into $4$ subsets of size $4$?

What is the number of partitions of a set of size $16$ into $4$ subsets of size $4$ ? What I have done: I have tried ${16 \choose 4} \cdot \frac{{12 \choose 4}}{4!}$ however this gives an answer of $...
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1answer
47 views

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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2answers
68 views

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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1answer
40 views

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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2answers
329 views

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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1answer
107 views

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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1answer
51 views

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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1answer
33 views

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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2answers
23 views

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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0answers
34 views

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 ...
2
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0answers
51 views

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 ...
3
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2answers
124 views

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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1answer
89 views

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....
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0answers
56 views

Negative multinomial expansion

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 ...
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2answers
100 views

How to find the coefficient of $x^{l}$ in the expansion of $(1+x+x^{2}+…+x^{n})^{m}$ for some given $l$? [duplicate]

Suppose I'd like to find the coefficient of $x^{l}$ in the expansion of $(1+x+x^{2}+...+x^{n})^{m}$, where $n$ and $m$ are given positive integers, for some given integer $l$ such that $n < l < ...
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1answer
22 views

Problem related to multinomial expansion.

How do I expand $(a_1+a_2+a_3+.....+a_k)^3$ where $a_i \in \Bbb R $ for $i = 1,2,...,k $?
4
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1answer
68 views

Evaluating the $n$th derivative of $(1+x+…+x^n)^d$ at $x=0$

I am trying to evaluate $$\frac{\mathrm{d}^n}{\mathrm{d}x^n}(1+x+x^2+...+x^n)^d\Bigr|_{x=0}.$$ If one starts deriving directly (or after summing the geometric series) the number of terms gets doubled ...
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1answer
56 views

How to calculate combination with 2 variables?

Combination formula with 1 variable is simple $nCr = n! / (r!(n-r)!)$ So to find a number of combinations for rolling a 6 ONCE out of 6 rolls is $$\dfrac {6! }{ (1!(6-1)!)} = 6$$ But how do you find ...

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