Rule of Product
The rule of product is central to counting. It is in fact so central that it's also known as the Fundamental Principle of Counting. It's pretty straightforward: if there are m ways to do something, and n ways to do something else, then there are \(m \cdot n\) ways to do both things at the same time.
Let's look at an example.
Example: Date night
Continuing with our date night from the previous section, a restaurant offers seven appetizers and eight main courses. How many possible meals are there?
Applying the rule of product, we have \(7 \cdot 8 = 56\) possible meals.
This rule extends to multiple variables in the obvious way.
Example: Date night with dessert
Yay, the restaurant offers dessert too. There's panna cotta, crème brûlée, tarts, shakes and more. In total there are ten dessert options. Now how many possible meals are there?
Applying the rule of product, we have \(7 \cdot 8 \cdot 10 = 560\) possible meals.
That's really all there is to it. Pretty easy, but foundational.
Exercise 1. Suppose a license plate has the following characteristics:
- The first three characters are letters in the range A-Z.
- The next three characters are numbers in the range 0-9.
- The last character is a letter in the range A-Z.
How many possible license plates are there?
Exercise 2. A room contains four on/off light switches. How many different ways are there to flip all four switches?