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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 $A$ , $B$ , and $C$ ), and $4$ pairs of pants (call them $w$ , $x$ , $y$ , and $z$ ). Then you have

$3 \times 4 = 12$

possible outfits:

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

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.