Rule of Product or Multiplication Principle

The Rule of Product, also known as the Multiplication Principle, is a basic counting principle that states that if there are ways to do one thing and ways to do another thing, then there are ways to do both things.

In other words, the total number of possible outcomes is the product of the number of possible outcomes of each event.

The Rule of Product can be used to solve many different types of counting problems, such as:

• How many different outfits can you create if you have 3 shirts and 2 pairs of pants?
• How many different ways can you roll a 6-sided die and then flip a coin?
• How many different license plates can be created if there are 26 letters and 10 numbers in the alphabet, and each license plate must have 6 characters?

Key Concepts

• Independent Events: The Rule of Product only works when the events are independent of each other. If one event depends on another, then the multiplication principle does not work.
• Two-Step Multiplication Principle: The Two-Step Multiplication Principle is used when a task can be broken up into two consecutive steps. If step 1 can be performed in m ways and for each of these, step 2 can be performed in n ways, then the task itself can be performed in m × n ways.

Examples :

Example 1:

You have 3 shirts and 2 pairs of pants. How many different outfits can you create?

Solution:

There are 3 ways to choose a shirt and 2 ways to choose a pair of pants, so there are different outfits possible.

Example 2:

You roll a 6-sided die and then flip a coin. How many different possible outcomes are there?

Solution:

There are 6 ways to roll the die and 2 ways to flip the coin, so there are different possible outcomes.

Example 3:

How many different license plates can be created if there are 26 letters and 10 numbers in the alphabet, and each license plate must have 6 characters?

Solution:

For the first character of the license plate, there are 26 letters to choose from. For the second character, there are 26 letters and 10 numbers to choose from, and so on.

Therefore, the total number of different license plates that can be created is .