Fundamental Principle of Counting (Part 3)


Hi. I hope that you now have some idea of the multiplication principle. This lesson will be focused on another basic principle of counting, known as the Addition Principle.

The Addition Principle

Let’s start with a simple problem: Suppose there are 3 different flights and two different trains connecting two places A and B. In how many ways can you reach from A to B (using only these flights or trains)?

permutations combinations

The answer seems obvious: 5

Why? Because you can either choose one of the flights (3 choices) or choose one of the trains (2 choices). Therefore the total number of choices are 3 + 2 = 5 (Flight 1 or Flight 2 or Flight 3 or Train 1 or Train 2)

Another one: A restaurant offers 4 non-veg dishes and 3 veg dishes as starters. How many different types of starters does the restaurant offer?

Again, the answer will be 4 + 3 = 7.

So what is the addition principle about? Here is one way to state it:

“If two events A1 and A2 can occur in m and n ways respectively (none of these being common), then either of these events can occur in m + n ways.”

An analogous statement for the multiplication principle will be:

“If two events A1 and A2 can occur in m and n ways respectively, then both of these events can occur together in m x n ways.”

Lesson Summary

The addition principle states that:

“If two events A1 and A2 can occur in m and n ways respectively (none of these being common), then either of these events can occur in m + n ways.”

In the next part of the lesson, we’ll see some cases when these principles cannot be applied directly, and how we can combine these two principles for such cases.


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