Fundamental Principle of Counting

Fundamental Principle of Counting Definition

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. 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. 

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 traditional method. (Say for the color red, the person can make 4 designs of tie for each shape. There are 3 shapes. So, 4 + 4 + 4 = 12 ties for red color 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 stich with the available combinations is calculated using 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 be 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 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 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:

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

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:

  • 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.

  • 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.

FAQ (Frequently Asked Questions)

1. What is the Fundamental Principle of Counting?

The fundamental principle of counting, also called the counting rule, is one of the ways to determine the total number of outcomes in probability examples. Generally, the number of ways in which the given events can happen individually is multiplied and the product indicates the total number of outcomes of all the given events. Basically, if an event A can happen in ‘c’ different ways and the event B can happen in ‘d’ different ways, then the events A and B can together happen in a * b different ways.

2. Give examples for the fundamental principle of counting.

Let us take a real life example of fundamental counting principle. Suppose we visit a hotel and order various kinds of dishes of our choice. Say for example, we get 2 kinds of soup and 4 types of salads as starters and later we are served with 7 kinds of main course dishes followed by 10 choices of beverages and 3 varieties of deserts. If we have to find the total number of ways in which the meals can be arranged as a combo, we use the fundamental principle of counting. We find the total number of ways in which the meal can be served with these varieties of dishes as:

2 x 4 x 7 x 10 x 3 = 1680 unique ways.