Counting for Fun, Basics – Part 2: The Rule of Product

The Rule of Product1 , Product Rule2 , Multiplication Rule3, or similarly named technique is used in counting problems often found in mathematics and computer science. Most all Discrete Mathematics textbooks cover this rule in the topic of counting. This rule serves as a basis for later counting techniques.

The Rule of Product
If there is a main task that can be broken down into two subtasks, and there are n ways of performing one of the subtasks, and there are m ways of performing the other subtask, then there are n · m possible ways to choose how to perform the main task.

Example 1
A parking lot is labeling their parking spaces with a single letter of the alphabet, and a single digit. How many parking spots will this labeling scheme cover?

The main task of labeling a parking spot consists of two substaks, picking one of the 26 letters of the alphabet and picking one of the 10 digits.

By the Rule of Product there are 26 · 10, or 260 parking spots that can be labeled.

Example 2
How many possible bit strings are there in a single byte?

There are 8 bits in a byte, and each bit has two states, 1 or 0. By the Rule of Product, there are 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2, or 2^8, or 256 possible bits strings in a byte.

Example 3
A state license plate labeling scheme is being updated. Each plate will have six characters. The first three characters may be either letters or digits. The remaining three characters may be digits. How many possible license plates can be produced using this labeling scheme?

The first, second, and third characters have 26 letter possibilities plus 10 digit possibilities, for a total of 36 possibilities for each character. The fourth, fifth, and sixth characters of the plate have 10 possible digits each. By the Rule of Product there are 36 · 36 · 36 · 10 · 10 · 10, or 36^3 \cdot 10^3, or 46,656,000 possible plates.


1. Wikipedia contributors. (2021, December 24). Rule of product. In Wikipedia, The Free Encyclopedia. Retrieved from
2. Rosen, K. H. (1999). Discrete mathematics and its applications. (4th ed., pp. 234). WCB/McGraw-Hill. ISBN 0-07-289905-0.
3. Epp, S. S. (2020). Discrete mathematics with applications.(5th ed., pp.575). Cengage. ISBN 978-1-337-69419-3.