A cup of warm soup.
Photo by Mick Haupt on Unsplash

Rule of Sum

Our first combinatorial principle is the rule of sum, which is also known as the addition principle. Intuitively, the rule states that if set \(A\) contains \(|A|\) choices, set \(B\) contains \(|B|\) choices, and the choices in \(A\) and \(B\) are mutually exclusive (i.e., you can't choose from \(A\) and \(B\) at the same time), then the number of choices is \(|A| + |B|\).

That is pretty abstract, but an example will make it easy to understand.

Example: Date night dinner

It's date night! At a restaurant, you get to choose an appetizer, which must be either a soup or a salad. There are three soups available and four salads available. Applying the rule of sum, the total number of choices is 3 + 4 = 7.

Unsurprisingly, this generalizes to n sets. If sets \(S_{1}, S_{2}, \ldots, S_{n}\) are pairwise disjoint (that is, \(S_{i} \cap S_{j} = \emptyset\) for \(i \neq j\)), then the total of choices is the sum of the number of choices for each of the individual sets. Symbolically:

$${|S_{1} \cup S_{2} \cup \ldots \cup S_{n}| = |S_{1}| + |S_{2}| + \ldots + |S_{n}|}$$

Another example will help.

Example: Date night movie

Time to pick a movie. The movie genres and corresponding movie counts are as follows:

Genre # Movies
Comedy 14
Drama 18
Action/Adventure 15
Horror 6
Science Fiction 5
Classic 8
International 2
Independent 4
Documentary 6

(For the sake of the example, assume that the genres are mutually exclusive. That is, no two genres include the same movie.) Then the total number of choices is

$${14 + 18 + 15 + 6 + 5 + 8 + 2 + 4 + 6 = 78}$$

Exercises

Exercise 1. At college, you are deciding whether to take Calculus 1A or else Introduction to Statistics course this semester. There are four course sections for calculus course and four for the stats course. How many course sections are there to choose from?

Exercise 2. You are the hiring manager for a certain position. Six of the candidates are male and eight candidates have a bachelor's degree. Can you use the rule of sum to determine how many candidates are either male or else have a bachelor's degree? Why or why not?