The Addition and Multiplication Principle Basic

Video tutorial

Lecture Notes

The addition principle and the multiplication principle are the two most fundamental ones in counting. They are so intuitive that many students might have already used them without knowing these two principles explicitly. However, having a thorough understanding will certainly improve one's ability to solve counting problems in a systematical way.

  • The addition principle can be applied when a counting problem involves multiple scenarios. The key is to ensure all the possible cases are covered with nothing missed or overlapped.
  • The multiplication principle can be applied when multiple independent steps are involved in completing one task. If steps are not independent, we cannot directly apply the multiplication principle (we will discuss this in the lesson Mind the Catches).

Tips: It is worth pointing out that when applying the multiplication principle, the sequence of chosen steps may be important. Certain  choices may lead to simpler solutions while others may lead to complex and tedious ones. We will see some examples in the lesson Mind the Catches later. If your solution is too convoluted, keep in mind that reordering your steps may simplify your solution.



Randomly draw a card twice with replacement from $1$ to $10$, inclusive. What is the probability that the product of these two cards is a multiple of $7$?

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Casework (also known as manual counting) is an application of the additional principle. To get a correct result, it is essential that all the possible cases are listed without duplication.

Note: There may be multiple methods to get to solve the same problem.

  • For the first example (# 4770), please ignore the last solution for now. That approach will be discussed later in the course Probability Basics.
  • The solution to the first assignment (# 1850) briefly mentions the "balls in the boxes" pattern. This can also be ignored. This pattern will be discussed later in the lesson Balls in Boxes.