# Statements > [!NOTE] Statement 1 (Product Rule for Counting) > Let $S$ and $T$ be [[Finite Set|finite sets]]. Then $\lvert S \times T \rvert=\lvert S \rvert \cdot \lvert T \rvert$ where $S\times T$ denotes their [[Cartesian Product|cartesian product]] and $|S|$ denotes the [[Cardinality|cardinality]] of $S.$ > [!NOTE] Statement 2 (General Product Rule/ Fundamental Counting Principle) > If we make a sequence of $m$ choices, where there are $k_{i}$ ways to make the $i$-th choice, then there $k_{1}\cdot k_{2}\cdot k_{3} \cdots k_{m}$ ways to make our sequence of choices. > > In other words, the number of ways to make a sequence of choices is the product of the number of options you have for each choice. # Proofs **Proof of equivalence of ....** .... $\blacksquare$ **Proof of statement 1.** .... **Proof of statement 2.** .... **Proof of statement 3.** .... # Applications **Consequences**: See [[Number of Lists of Length k whose Elements are Taken From a Finite Set]] # Bibliography 1. Bogart, Kenneth P. _Combinatorics through Guided Discovery_. 2004. Open Access Textbooks, www.math.dartmouth.edu/~kbogart/pub/guided-discovery.pdf. 2. [[ST120 Introduction to Probability]].