When dealing with the occurrence of more than one event or activity, it is important to be able to quickly determine how many
possible outcomes exist.
For example, if ice cream sundaes
come in 5 flavors with 4 possible toppings, how many different
sundaes can be made with one flavor of ice cream and one
Rather than list the entire sample space
with all possible combinations of ice cream and toppings, we may simply
multiply: 5 • 4 = 20 possible sundaes. This simple
multiplication process is known as the
The Fundamental Counting
Principle: If there are
a ways for
one activity to occur, and b
ways for a second activity to occur, then there are
a • b ways for both to
1. Activities: roll a die and flip
There are 6 ways to roll a die and 2 ways to flip
There are 6 • 2 = 12 ways to roll a die and flip
Activities: draw two cards from a standard deck of 52 cards without
replacing the cards
There are 52 ways to draw the first card.
There are 51 ways to draw the second card.
There are 52 • 51 = 2,652 ways to draw the two
The Counting Principle
also works for more than two activities.
a coin is tossed five times
There are 2 ways to flip each coin.
There are 2 • 2 • 2 • 2 •2 = 32
arrangements of heads and tails.
a die is rolled four times
There are 6 ways to roll each die.
There are 6 • 6 • 6 • 6 = 1,296 possible
The Counting Principle is easy! Simply MULTIPLY the
number of ways each activity can occur.