T1: More than Counting

May 06, 2024

• A Small Problem
• A Higher Binomial Theorem

A Small Problem

$\textbf{Question 1.}$ If you have $n$ elements you wish to divide into $y$ distinct piles of size $n_1,n_2,\ldots,n_y$, how many possibilities?

The answer is

$${n \choose {n_1,n_2,\ldots,n_y }}= \frac{n!}{n_1!n_2!\ldots,n_y!}$$

A Higher Binomial Theorem

$\textbf{Multinomial Formula}$. For all $n,y \in \mathbb{N}$, for all pairwise commuting variables $x_1,\ldots,x_y$, we have $(x_1+x_2+\ldots+x_y)^n = \sum_{n_1,\ldots,n_y: \sum_{i=1}^y n_i = n}{n \choose {n_1,\ldots,n_y}}x_1^{n_1}x_2^{n_2}\ldots x_y^{n_y}$