2.3 Basic Counting Principles

Addition

The addition principle of counting is a basic idea for counting the total number of possibilities that can occur when dealing with many events or options that are mutually exclusive. The events that are mutually exclusive are those that cannot occur simultaneously. This rule is used to get the total number of ways that an event can occur. It is typically used when you have to make a choice between multiple option.

The Addition principle is written as:

For example, a student wants to buy a new computer. He has to decide among 3 desktop computers and 4 laptop computers.

Total number of computer options $=$ Total options in desktop computers $^+$ Total options in laptop computers

So in this case

Multiplication

The Multiplication Principle is used when we are dealing with many events, and we have to select one option from each event.

For example: Bilal has 3 Apples, 3 Bananas and 3 bunch of Cherries. In how many ways, he can make fruit basket by putting fruit of one kind.

(A1 B1 C1) (A1 B1 C2) (A1 B1 C3) (A1 B2 C1) (A1 B2 C2) (A1 B2 C3) (A1 B3 C1) (A1 B3 C2) (A1 B3 C3) (A2 B1 C1) (A2 B1 C2) (A2 B1 C3) (A2 B2 C1) (A2 B2 C2) (A2 B2 C3) (A2 B3 C1) (A2 B3 C2) (A2 B3 C3) (A3 B1 C1) (A3 B1 C2) (A3 B1 C3) (A3 B2 C1) (A3 B2 C2) (A3 B2 C3) (A3 B3 C1) (A3 B3 C2) (A3 B3 C3)

Using Counting Principle Problems, the total number of ways of choosing this pairing would be:

Options available for Apples $(A) = 3$ Options available for Bananas $(B) = 3$ Options available for Bunch of Cherries $① = 3$ Total no.of ways: $3\times 3\times 3=27$$3\times 3\times 3 = 27$

Permutation

Many different kinds of problems can be resolved using the Multiplication Principle. Putting objects in order is one kind of challenge. We queue up for pictures, arrange letters into words and digits into numbers, decorate rooms, and more. A permutation is a way to arrange items.

Drawing line segments for each choice can be useful in solving permutation problems. This allows us to multiply by figuring out how many of each option there are.

Assume we have four paintings and three places on the wall to display. We wish to determine how many different ways there are to arrange three of them on a wall. To symbolize the three locations on the wall, we can draw three lines.