# [Maths Class Notes] on Fundamental Principle of Counting Pdf for Exam

The term ‘counting’ is the fundamental concept of Mathematics. The whole world of Mathematics started with the basic necessity of counting. Our ancestors first used fingers for counting and later started using beans, sticks, buttons, and beads to count. However, they, later on, realized that these methods of counting cannot be used in cases where we are forced to count large and large quantities of numbers. That is when our Mathematicians came out with a way of determining large counts efficiently and accurately with the help of the fundamental counting principle. The fundamental counting principle is one of the most important rules in Mathematics especially in probability problems and is used to find the number of ways in which the combination of several events can occur.

Mathematics is an interesting subject, here every concept has a different technique and method of playing with numbers. The application of Mathematics is vast and touches every discipline in some or another way. To make the learning more fun and engaging for the students, the subject experts at have come up with unique study sessions where new concepts are being taught. Students can either read the articles on a wide range of topics or can watch the video lectures on any topic.

In this particular article, we aim at making you learn one such important concept of Mathematics.

Table of Content –

Let’s begin learning together.

### Explanation of Fundamental Principle of Counting Definition

Consider an example of a person who runs a business of sewing neckties. He can make ties to be unique based on the following factors: color, shape, and design. Suppose he has a choice of 5 colors, 3 shapes, and 4 different design patterns. To find the number of unique ties he can make, it becomes a complex calculation if we are counting by the traditional method. (Say for the color red, the person can make 4 designs of a tie for each shape. There are 3 shapes. So, 4 + 4 + 4 = 12 ties for red alone.) The traditional method may seem to be easy for smaller numbers. However, it is not as easy as imagined for a larger number of outcomes for each event. So, in this case, the number of ties the person can stick with the available combinations is calculated using the Fundamental Principle of Counting definition as:

Total number of unique ties = 5 x 3 x 4 = 60

This method of multiplication can be employed in solving the probability problems wherever there are different kinds of events taking place at the same time. The fundamental rule can be used over a set of categories when one or more out of several choices in each of the categories is to have opted.

### A Deeper Understanding of the Fundamental Counting Principle

The fundamental counting principle or basic principle of counting is a method or a rule used to calculate the total number of outcomes when two or more events are occurring together. This principle states that the total number of outcomes of two or more independent events is the product of the number of outcomes of each individual event. For example, a child choosing among six flavors of icecreams with 3 varieties of cones will have 6 x 3 = 18 different choices of icecreams.

### Fundamental Counting Principle Examples in Real Life

1. A boy has 4 T-shirts and 3 pairs of pants. Find the total number of possible outfits the boy has.

Solution:

The above question is one of the fundamental counting principle examples in real life.

According to the question, the boy has 4 t-shirts and 3 pairs of pants.

So, the total number of outfits with the boy are:

Total number of outfits = 4 x 3 = 12

The boy has 12 outfits with him.

1. Consider an example where a fair die is rolled and a card is drawn from a deck. What is the total number of outcomes in this case?

Solution:

A total number of outcomes can be found by considering the above example as one of the fundamental counting principle examples in real life.

A total number of outcomes can be calculated as the product of the number of outcomes when a die is rolled and the number of outcomes when a card is drawn from the deck.

If the number of outcomes of a rolled die is ‘p’ and that of the card being drawn from the deck is ‘q’, then the total number of outcomes is calculated as p x q.

A fair die has six faces. So the total number of outcomes in case of a die is p = 6.

A deck of cards has 52 cards. So, the total number of possible outcomes when a card is drawn is q = 52.

So, the total number of outcomes when both the events occur at the same time is:

p x  q = 6 x 52 = 312.

### Fun Facts about Fundamental Principle of Counting

• The Fundamental Principle of Counting can be extended to the examples where more than 2 choices are there. If an event can happen in ‘x’ ways, the other event in ‘y’ ways, and another one in ‘z’ ways, then there are x * y * z ways for all the three events to happen.

• The fundamental counting principle is also called the Counting Rule.

• If the same number of choices repeat in several slots of a given fundamental counting principle example, then the concept of exponents can be used to find the answer.