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. 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 Vedantu 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 -
An introduction
Definition Fundamental Principle of Counting
Real-life examples
Fun Facts
Frequently asked question
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
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.
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.
FAQs on Fundamental Principle of Counting
1. How is permutation different from the Fundamental Principle of Counting?
The two may find some similarities but the Fundamental Principle of both concepts are different. The Fundamental Principle of Counting is to find the total number of outcomes when two or more events are occurring together. For say if there are X ways of doing a task and Y ways of doing another, then there are X*Y ways of doing both the tasks. On the other hand, Permutation is based on the principle of arranging an object without repetition.
2. What are some of the everyday applications of the Fundamental Principle of Counting?
The concept can be used in multiple ways -
If you have n numbers of dishes you can find out the ways in which they can be presented
Counting helps you know the number of events that can occur and thus help you make the decision
the Fundamental Principle of Counting is widely used in statistics and data analysis
You can think of more such events which have more than two choices, there we can apply the concept of the Fundamental Principle of Counting.
3. What do you mean by the term sample space in the Fundamental Principle of Counting?
Sample space is the complete set of possible outcomes that can occur in a single event. For example, if you flip a coin then it has the probability of displaying heads or tails. Here, head and tail is the sample space for the event of flipping a coin because these are the only probability they can have. So, the event is the wider concept which has sample space, and finally, there is the sample point which is the final outcome of the single event.
Students can watch the video lecture to know more about the concept.
4. Is the topic Fundamental Principle of Counting difficult to understand?
The faculties at Vedantu understand the difficulty many students undergo while understanding the concepts of Math. We have explained the topic using the most simple language and also explained it with the help of multiple examples. The topic has been simplified to the extent that students can learn the concepts through self-reading.
Also, the subject experts are available for the help of the students making them comfortable with the subjects.
Irrespective of all the support provided by the team, the topic is simple on its own.
5. If we can count things manually why do we need the Fundamental counting principle?
See, the process of counting is age-old. People used to count on fingers or through tally counting. But as the system started getting complex and the items to count increased, it became impossible to count through the conventional methods. To meet the growing demand the Fundamental counting principle was introduced, it can make it possible to count any number of events and combine with ease.
Understand the concept and try to count events around you using these principles.
6. 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.
7. 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.
