Articles of multinomial coefficients

Calculation of covariances $cov(x_i^{2},x_j)$ and $cov(x_i^{2},x_j^{2})$ for multinomial distribution

We know that,$ \ \ \ \ cov(x_i,x_j)=-n \ x_i \ x_j$. It can be proven in this manner: We know, $Var(x_i+x_j)=cov((x_i+x_j),(x_i+x_j))$ Now, $cov((x_i+x_j),(x_i+x_j))=cov(x_i,x_i)+2 \ cov(x_i,x_j) + cov(x_j,x_j)=Var(x_i)+Var(x_j)+2 \ cov(x_i,x_j)$ Since, $Var(x_i+x_j)=n(p_i+p_j)(1-p_i-p_j)$ and $Var(x_i)=np_i$ and $Var(x_j)=np_j$ Hence, $cov(x_i,x_j)=(\frac{1}{2})[Var(x_i+x_j)-Var(x_i)-Var(x_j)]=(\frac{1}{2})[n(p_i+p_j)(1-p_i-p_j)-np_i-np_j]=(\frac{1}{2})[-2 \ n \ p_i p_j]=-n p_i p_j$ Hence, $cov(x_i,x_j)=-n \ p_i \ p_j$ Now I am interested […]

How many summands are there

I have some problem understanding this Exercise/problem. What is summand ? I have searched for it, but nothing concrete came up. Problem: Look at the multinomial theorem. How many summands are there in $(x+y+z)^7$ and in $(w+x+2y+z)^9$ ? Can someone explain to me what it is(summand) and how to get the solution. The solution should […]

Multinomial Theorem Example Questions

I’m learning about the multinomial theorem and working 2 examples in a book. I thought I understood the examples until I did example 5c. I don’t understand why these two examples are different. In example 5c it says that order is now irrelevant. What do you mean order is irrelevant? Why was the order relevant […]

Divisibility of multinomial by a prime number

What is the condition for divisibility of multinomial $ \dbinom {n}{x_1, x_2, \dots, x_k} $ by a prime $p$? Update: I tried to solve using a generalisation of Lucas Theorem by representing the $n$ as well as $x_1, x_2,\dots,x_k$ in base $p$ representation. I am following this site – Editorial to a problem I am […]

Hockey-Stick Theorem for Multinomial Coefficients

Pascal’s triangle has this famous hockey stick identity. $$ \binom{n+k+1}{k}=\sum_{j=0}^k \binom{n+j}{j}$$ Wonder what would be the form for multinomial coefficients?

2D walks on a square grid; The number of Paths leading to specific $(X,Y)$

Introduction (1) Lets have a 2D plane, and place a Walker in the center $(X,Y)=(0,0)$ Lets take a example where we use all of the possible moves, $(m = 9)$. We have one such case, where the Walker can make one of the $9$ moves each turn: Up, Down, Left, Right, Up-right, Down-right, Up-left, Down-left […]

Finding the coefficient using the multinomial theorem?

Set $F := F (X, Y, Z) = (X^2 + 3Y − Z^2)^8$. Determine the coefficients with which the following terms appear in $F$. $X^4 Y^2 Z^2.$ $X^{10} Y^2 Z^2$. I would know how to find the coefficient if it was just $(X^2+Y-Z^2)^8$. It is the coefficient of the $y$ term that is making me […]

Series expansion of infinite series raised to the $n$th power

So I know there is a well-known straightforward way to expand something like $$(a+b)^n$$ and that there are formulas which allow us to expand trinomials and multinomials in general. My question is, Is there any known way to expand something like $$\left[\sum_{k=0}^{\infty} a_k\right]^n$$ or at least to determine the first few terms?

Challenge: How to prove this identity between bi- and trinomial coefficients?

This question is the continuation of its predecessor. Using the convention that trinomial coefficients $$ \binom{n}{k_1,k_2,k_3}=\frac{n!}{k_1! k_2! k_3!} $$ are zero if $k_i<0$ or $\sum_i k_i\neq n$, we have the following relation between binomial and trinomial coefficients, $$ \binom{2n}{\ell} = \sum_{k=0}^{\lfloor \frac{\ell}2 \rfloor} 2^{\ell-2k} \binom{n}{\ell-2k,k,n-\ell+k} = \sum_{k=\max(\ell-n,0)}^{\lfloor \frac{\ell}2 \rfloor} \frac{2^{\ell-2k} n!}{(\ell-2k)!\,k!\,(n-\ell+k)!}, \quad (*) $$ and […]

What are the coefficients of the polynomial inductively defined as $f_1=(x-2)^2\,\,\,;\,\,\,f_{n+1}=(f_n-2)^2$?

Let $\{f_n(x)\}_{n\in \Bbb N}$ be a sequence of polynomials defined inductively as $$\begin{matrix} f_1(x) & = & (x-2)^2 & \\ f_{n+1}(x)& = & (f_n(x)-2)^2, &\text{ for all }n\ge 1.\end{matrix}$$ Let $a_n$ and $b_n$ respectively denote the constant term and the coefficient of $x$ in $f_n(x)$. Then $(\text A)\,\, a_n=4, b_n=-4^n$ $(\text B)\,\,a_n=4, b_n=-4n^2$ $(\text C)\,\, […]