Addition method

Addition method (also called the sum rule) is one of the most fundamental counting techniques. It is used when we have to choose either one thing or another, but not both at the same time.

 

The method is based on the idea that if we have two or more options that do not overlap, we can find the total number of possibilities by adding them together.

 

 

Example 1: Transport

A student has to choose transport to school:

 

• Either: Cycle (1 option)
• Or: Take the bus (2 different lines)
• Or: Take the train (1 option)

 

Since this is a choice of either cycle or bus or train, we must add the possibilities together:

 

$$ \large 1 + 2 + 1 = 4\ options $$

 

 

Example 2: Ice cream flavors

An ice cream shop offers two types of cones:

 

• Cone A: 3 different scoops to choose from
• Cone B: 4 different scoops to choose from

 

If you choose either Cone A or Cone B, the total number of options is:

 

$$ \large 3 + 4 = 7\ options $$

 

 

Formula

In general:

 

$$ \large \text{Number of possibilities} = a + b + c + \ldots + n $$

 

 

When can you use the Addition method?

 

• When it is a case of mutually exclusive choices. You can only choose one option at a time.
• When none of the options overlap. Otherwise, you risk double-counting.

 

A simple diagram with three branches (cycle, bus, train) can illustrate the method. The total number of branches corresponds to the sum of the options.

 

 

When options overlap

The Addition method only works if the options are completely separate. If there is overlap, we cannot just add them, because we would otherwise count the overlapping options twice.

 

Example: A bookstore sells math books and physics books. There are 20 math books and 15 physics books. Five of the books belong to both subjects. If we just added the numbers, we would get:

 

$$ \large 20 + 15 = 35 $$

 

But the correct number is:

 

$$ \large 20 + 15 - 5 = 30 $$

 

For this type of problem we instead use the inclusion–exclusion principle, which is an extension of the addition method.

 

 

Summary

The addition method is used when you face a choice where you can choose either one option or another. It gives you the total number of possibilities by simply adding them together.

 

The method is therefore a foundation in many problems in combinatorics.

If the options overlap, we must move on to the inclusion–exclusion principle, which corrects for double-counting.