The Counting Principle
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 topping?
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 Counting Principle.
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 occur.
Examples:
1. Activities: roll a die and flip a coin
There are 6 ways to roll a die and 2 ways to flip a coin.
There are 6 • 2 = 12 ways to roll a die and flip a coin.
2. 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 cards.
The Counting Principle also works for more than two activities.
3. 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.
4. Activities: a die is rolled four times
There are 6 ways to roll each die.
There are 6 • 6 • 6 • 6 = 1,296 possible outcomes.