Questions tagged [multinomial-coefficients]
For questions related to multinomial coefficients, a generalization of binomial coefficients.
358
questions
0
votes
1answer
32 views
Trinomial equation - combinatorial explanation
Assume that $e+f+g=n$.
Prove:
$P(n;e,f,g)=P(n-1;e-1,f,g) + P(n-1;e,f-1,g) + P(n-1;e,f,g-1)$
Where $P(n;e,f,g)$ is the tri-nomial-coefficient of n over e,f,g ($\frac{n!}{e!f!g!}$)
Combinatorially.
2
votes
3answers
80 views
bounding extended binomial coefficients from above
Given natural $i,m\ge 1$, how large can the largest coefficient of the polynomial $$(x^0+x^1+\dots+x^{m-1})^i$$ (viewed as polynomial in $x$) be?
A trivial upper bound is $m^i$, perhaps even $m^{i-1}$....
0
votes
1answer
22 views
Class of 16 Participants Answering 6 Questions in Subsets of 4, Each One in A Different Combination With All Pairings Covered
16 Participants are arranged into 4 groups of 4. The participants work together on a question within their groups. Next the groups are rearranged into another 4 groups of 4, where they work on the ...
1
vote
0answers
32 views
Trinomial Expansion and Complex Numbers
Consider the expression $$(1 + x + x^2)^n = A_0 + A_1x + A_2x^2+\cdots + A_{2n-1}x^{2n-1} + A_{2n}x^{2n}$$ (where $n$ belongs to positive integers).
$n\equiv 0\mod 4 \implies \quad A_0 - A_2 + A_4 - ...
2
votes
1answer
525 views
Trinomial Theorem and Complex numbers
Consider the expression $$(1 + x + x^2)^n = C_0 + C_1x + C_2x^2+\cdots + C_{2n-1}x^{2n-1} + C_{2n}x^{2n}$$ (where $n$ belongs to positive integers),then
the value of
$C_0 + C_3 + C_6 + C_9 + C_{12}+\...
1
vote
1answer
21 views
Number of outcomes in permutation invariant Multinomial Distributions
I have a dice with $K$ many outcomes. I am rolling this dice $n$ times. Assume $k_i$ denotes number of times class $i = 1,\ldots,K$ appears in $n$ many rolls. I am wondering, in how many different ...
0
votes
1answer
19 views
Assign the number of ways to share 16 identical objects
In how many different ways can be shared 16 identical objects to 7 different persons such that 3 of them can accept maximum of 2 objects, 3 of them at least 2 objects and for the other person don't ...
0
votes
3answers
76 views
Find the coefficient of $x^k$
Find the coefficient of $x^k$ in $$\frac{1}{(1+x)(2-9x)}.$$
This problem is in chapter of Algebraic Tools, using generating functions.
0
votes
0answers
15 views
Polynomials and mulinomial coefficients [duplicate]
Let it be $$(1+x+x^2+x^3+x^4)^{496}=a_0+a_1x+a_2x^2+...+a_{1984}x^{1984}.$$
Find $$\gcd(a_3,a_8,...,a_{1983})$$
I know the Leibniz Theorem, so all coefficients are of the form
$$\frac{496!}{a!b!c!d!e!}...
0
votes
0answers
17 views
Trinomial Summation Equation [duplicate]
I was looking at how to find trinomial coefficients with code and I was puzzled with this equation.
$$\binom n k _2 = \binom{n}{-k}_2$$
I know that (nCk) represents n choose k but what does the 2 at ...
0
votes
1answer
34 views
Polynomials and Number Theory
Given the equation
$(1+x+x^2+x^3+x^4)^{496}=a_0+a_1x+a_2x^2+...+a_{1984}x^{1984}$
a) What is the gcd of $(a_1,a_2,a_3,...
a_{1983})$?
b) Show that $10^{340}< a_{992}< 10^{347}$.
By the ...
2
votes
2answers
42 views
Multinomial Coefficient of $x^{1397}$ in expansion of $(x^3+x^4+x^5+…)^6$
I have the following problem:
Find the Coefficient of $x^{1397}$ in expansion of $(x^3+x^4+x^5+...)^6$
I know how to solve these kind of questions using Multinomial Theorem but since the polynomial ...
1
vote
0answers
22 views
Understanding coefficients of related polynomials
Im trying to understand this concept in general but as an example lets say I have a polynomial like:
$$f=(1+x+x^2)(1+x+2+(x+2)^2+(x+2)^3+(x+2)^4)(1+x+6+(x+6)^2 $$ which can simplify to:
$$ f= (x^8 + ...
0
votes
0answers
11 views
Monotonicity of a probability that is related to a multinomial distribution.
Consider a multinomial distribution with three outcomes. Let $x_i$ denote the number of occurences of the $i^{th}$ outcome, and the $i^{th}$ outcome occurs with probability $p_i$, $i=1,2,3$. Let $n$ ...
0
votes
2answers
55 views
Intuitive meaning for multinomial coefficient. (Why only one?) [closed]
Introduction
I am doing a (university) probability course, wherein there seem to be occasional errors in what is taught. [Edit: this is not just my opinion.]
<...
6
votes
1answer
50 views
Sum of multinomial coefficients with bounded indices
I want to know if there's a closed formula for
$$
f_{k}(n_{1}, \dots, n_{k}) = \sum_{i_{1} = 0}^{n_{1}} \cdots \sum_{i_{k} = 0}^{n_{k}} \frac{(i_{1} + \cdots + i_{k})!}{i_{1}!\cdots i_{k}!}
$$
for $k\...
0
votes
2answers
31 views
Binomial coefficient after expansion
I am trying to solve an exercise where in the final step, I need to find the coefficient of $x^7y^5$ in $(x+y)^{12} + 7(x^2+y^2)^6 + 2(x^3+y^3)^4 + 2(x^4+y^4)^3 + 2(x^6+y^6)^2 + 4(x^{12}+y^{12}) + 6(x+...
3
votes
1answer
137 views
A certain composition into the elementary symmetric polynomials
Preliminaries
Let $\mathbb{F}$ be a field such that $\operatorname{char}(\mathbb{F})\neq2$.
Let $n$ be a non-zero natural number. Let $\mathbb{F}\left[x_1,x_2,\ldots,x_n \right]$ be a polynomial ...
4
votes
2answers
120 views
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: ...
0
votes
1answer
39 views
Prove $\binom{n}{k_1,…,k_m} = \sum_{i=1}^m \binom{n - 1}{k_1,…,k_{i - 1},..,k_m}$
I have a question ask to prove $$\binom{n}{k_1,...,k_m} = \sum_{i= 1}^m \binom{n - 1}{k_1,...,k_{i - 1},..,k_m}$$
I'm not sure how to approach this question, but the only thing that I noticed the LHS ...
0
votes
2answers
57 views
Given $(x+y+z)^{15}$ find the coefficient of $x^2y^{10}z^{3}$ [duplicate]
Given $(x+y+z)^{15}$ find the coefficient of $x^2y^{10}z^3$
Weak in this chapter. Don't know how to proceed. Please help
Sorry for the typo made before
1
vote
1answer
52 views
The maximal term of the multinomial distribution
(Feller volume 1, Q28, p.171) Suppose that we have the following binomial distribution
$$\frac{n!}{k_1!k_2!... k_r!} p_1^{k_1}p_2^{k_2} ... p_r^{k_r}.$$
Prove the theorem. The maximal term of the ...
0
votes
0answers
16 views
Is there a way to find the coefficients of this polynomial without expanding it?
If the expression $(x^3-x^2y+xy^2+y^3)^3$ is expanded and simplified, what is the sum of all the coefficients of the resulting polynomial?
I know the answer is 8, and I think last time I did it I ...
1
vote
2answers
99 views
counting allowable/admissible paths on a grid with obstacles
I have a grid where some paths are removed.
where the following path is admissible,
but this path
is not.
How can I go about finding how many admissible paths are there on the following 2 grids (...
0
votes
2answers
63 views
calculating total number of allowable paths
I seem to be struggling with the following type of path questions
Consider paths starting at $(0, 0)$ with allowable steps
(i) from $(x,y)$ to $(x+1,y+2)$,
(ii) from $(x,y)$ to $(x+2,y+1)$,
(iii)...
0
votes
0answers
18 views
greatest term in multinomial expansion proof
Can anyone give the proof for greatest term in multinomial expansion (coefficients of x are not 1)and an intuitive sense for the formula.Our book just contains the formula without any understanding of ...
1
vote
1answer
31 views
calculating allowable paths
I have the following question, usually I'd solve by a modification of pascals triangle but I'm not sure how to approach this using pascals since step D is problematic. How could I go about this?
...
2
votes
4answers
74 views
Finding a coefficient of $x^{57}$ in a polynomial $(x^2+x^7+x^9)^{20}$
So the task is to find a coefficient of $x^{57}$ in a polynomial $(x^2+x^7+x^9)^{20}$
I was wondering if there is a more intelligible and less exhausting strategy in finding the coefficient, other ...
4
votes
2answers
84 views
Multinomial Distribution — How to calculate percentiles?
I've read the rules and searched but I do not even know what I'm looking for.
Here is my problem:
Suppose I have a bag containing three different marbles: red, green, and blue. I am drawing a single ...
3
votes
3answers
79 views
calculating total number of allowable paths from $(0,0)$ to $(5,5)$
I'm looking at paths starting from $(0,0)$ with the following allowable steps :
1) from $(x,y)$ to $(x,y+1)$
2) from $(x,y)$ to $(x+1,y)$
3) from $(x,y)$ to $(x+2,y+1)$
how can I determine the ...
2
votes
2answers
69 views
How many ways to form a 4 letter word when order doesn't matter and letters need not be different
Can someone please explain this. How many 4 letter words are there, when order doesn't matter and letters can be repeated ?
IF I do in one approach I get $\frac{26^4}{4!} $ (26 letters for each ...
0
votes
2answers
68 views
In the multinomial expansion of $(a+b+c)^5$ , why does “a” have $5 \choose 1$ positions? [closed]
Also, why does b has $4\choose 2$ positions and c has $3 \choose 2$ positions? , i.e. $(a+b+c)^5=a^5+...+abbcc+babcc+cabcb+....$
1
vote
1answer
87 views
What is the math behind this Python program generating multinomial coefficients?
I wrote a Python program that is using recursion to generate multinomial coefficients - see next section.
Mathematically it is also using recursion by 'decrementing down to the boundary'.
My ...
0
votes
0answers
38 views
Combinatorics and Multinomials: How to find the number of distinct terms?
How do I find the number of distinct terms in (a - 2b + 3c + d)āæ where n = 17? I started the problem by realizing that the multinomial expansion says that to find the number of distinct terms for (a + ...
0
votes
1answer
33 views
Algebraic identity for summing all arrangements of n choose k terms
As the title suggests, I am looking for this type of identity, where k is some integer between 3 and n-2. I know, for example, that $$\frac{(x_1^2+x_2^2+...+x_n^2)-x_1^2-x_2^2-...-x_n^2}{2}=x_1x_2+...
2
votes
4answers
121 views
Sum of coefficients of $x^i$ (Multinomial theorem application)
A polynomial in $x$ is defined by
$$a_0+a_1x+a_2x^2+ \cdots + a_{2n}x^{2n}=(x+2x^2+ \cdots +nx^n)^2.$$
Show that the sum of all $a_i$, for $i\in\{n+1,n+2, \ldots , 2n\}$, is
$$ \frac {n(n+1)(5n^...
0
votes
0answers
48 views
Approximation of multinomial distribution for large N (Expected end-to-end distance of a random walk on a hypercubic lattice in arbitrary dimension)
I was trying to derive an expression for the expected end-to-end distance of a random walk on a hypercubic lattice with sides of length $l$ in an arbitrary dimension $d$ after $N$ steps.
I found that ...
0
votes
0answers
16 views
1
vote
1answer
68 views
Coefficient shortcut
I'm taking the product of a specific set of polynomials of idempotent variables and want to find a shortcut for counting groups of homogeneous coefficients without performing all the calculations.
...
1
vote
2answers
94 views
Can a binominal (or multinomial) coefficient be computed efficiently?
It would seem that a preceding query would be on point, but not really for me. One of the answers comes close, but it isn't complete as is. Since the answer is going to be an integer, all the ...
1
vote
2answers
79 views
What's the coefficent of $x^{20}$ in $(x^3+x^4+x^5+…)^5$? Only hint is needed.
I know about Binomial/Multinomial expansion but I got stuck on this series; it doesn't look like anything I've solved before. I already searched for any hint/formula and couldn't find one, any help is ...
0
votes
1answer
21 views
A Conceptual doubt on Multinomial Theorem.
I was solving some questions of Multinomial Theorem from Higher Algebra by Hall and Knight when I encountered this question.
$\text{If }\left(1+x+x^{2}+\ldots+x^{p}\right)^{n}=a_{0}+a_{1} x+a_{2} x^...
1
vote
2answers
35 views
A question on the multinomial expansion
Given that $$(2x^2+3x+4)^{10}=\sum_{i=0}^{20} a_{i}x^{i} $$
Calculate the value of $\frac {a_{7}}{a_{13}} $.
I have manually taken all the cases of formation of $x^7$ and $x^{13}$ and arrived at the ...
0
votes
1answer
27 views
multinomial sum upper bound
I am looking for an upper bound on the following sum, where $p_1,\dots,p_r > 0$ and $\sum_i p_i\leq 1$.
$$
\sum_{0\leq i_1,\dots,i_r\leq n} \binom{i_1+\cdots+i_r}{i_1,\dots,i_r} p_{1}^{i_1}\cdots ...
0
votes
4answers
71 views
Find coefficient in cubic polynomial
We have
$$ p(x) = ax^3 + bx^2 + cx +d$$
where $a, b, c, d$ are complex coefficients. We have to find all posible coefficients for:
$$ p(1) = 2$$
$$ p(i) = i$$
$$ p(-1) = 0$$
Unfortunately I dont know ...
0
votes
1answer
30 views
How to compute the coefficients (ODE using power series method)?
How to Solve $xy'+2y=4x^2;y(1)=5$ using power series method?
I've assumed $y = \sum_{m=0}^\infty a_mx^m\implies y' = \sum_{m=1}^\infty m a_mx^{m-1}$.
Putting these values in equation in $xy'+2y=4x^...
2
votes
3answers
94 views
Find coefficient for $x^{10}$ in $x^3(x^2-3x^3-1)^6$
I'm trying to solve the following problem:
Find the coefficient for $x^{10}$ in $x^3(x^2-3x^3-1)^6$.
Can I use the multinomial theorem to solve it? I'm unsure how to start..
Thanks!
2
votes
1answer
41 views
How to prove that ${n \choose r_{1},\ldots,r_{k}}=\sum_{j=1}^{k}{n-1 \choose r_{1},\ldots,r_{j}-1,\ldots,r_{k}}$ using algebra?
Im trying to prove using algebra that
$$
{n \choose r_{1},\ldots,r_{k}}=\sum_{j=1}^{k}{n-1 \choose r_{1},\ldots,r_{j}-1,\ldots,r_{k}}
$$
Attempt:
The multinomial theorem states that
$$
\left(x_{1}+\...
1
vote
2answers
186 views
Counting paths in multidimensional grids
Let's consider a grid in a multidimensional space. Each grid has 'n' number of lines, spaced one unit apart. A path (or walk) of 'm' steps is between two grid intersections. A step is a single unit ...
1
vote
1answer
41 views
Coefficient of $x^4y^3z^3$ in the expansion of $(5x+y-4z)^{10}$
The coefficient of $x^6y^4$ in the expansion of $(2x-3y)^{10}$ is
$$_{10}C_6 \cdot 2^6 \cdot (-3)^4$$
and as for the coefficient of $x^3y^4z^8$ in the expansion of $(x+y+z)^{15}$ is
$$_{15}C_3 \...
