# Fundamental Counting Principle

The
**
fundamental counting principle
**
states that if there are
$p$
ways to do one thing, and
$q$
ways to do another thing, then there are
$p\times q$
ways to do both things.

**
Example 1:
**

Suppose you have $3$ shirts (call them $\text{A}$ , $\text{B}$ , and $\text{C}$ ), and $4$ pairs of pants (call them $w$ , $x$ , $y$ , and $z$ ). Then you have

$3\times 4=12$

possible outfits:

$\begin{array}{l}\text{A}w,\text{A}x,\text{A}y,\text{A}z\\ \text{B}w,\text{B}x,\text{B}y,\text{B}z\\ \text{C}w,\text{C}x,\text{C}y,\text{C}z\end{array}$

**
Example 2:
**

Suppose you roll a $6$ -sided die and draw a card from a deck of $52$ cards. There are $6$ possible outcomes on the die, and $52$ possible outcomes from the deck of cards. So, there are a total of

$6\times 52=312$

possible outcomes of the experiment.

The counting principle can be extended to situations where you have more than $2$ choices. For instance, if there are $p$ ways to do one thing, $q$ ways to a second thing, and $r$ ways to do a third thing, then there are $p\times q\times r$ ways to do all three things.

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