Math Homework. Do It Faster, Learn It Better.

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 × 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 × 4 = 12

possible outfits:

A w , A x , A y , A z B w , B x , B y , B z C w , C x , C y , C z

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 × 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 × q × r ways to do all three things.