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When we start dealing with mathematics the basic concept we learn is counting. If you don’t know how to count you can’t learn any other concept in mathematics so, we can say that counting constitutes the whole of mathematics. Mathematics has a wide range of applications and affects almost every discipline in some way. If we talk about permutation and combination the knowledge of the fundamental principle of counting is really important. As we have to find a number of ways in which we can select or arrange objects. Even if we talk about probability then knowledge of the fundamental principle of counting is important. The basic rule of counting the total number of possible outcomes in a situation is known as the fundamental counting principle. We have two basic fundamentals of counting i.e. multiplication and addition.
What is the fundamental principle of counting?
The fundamental counting principle, sometimes known as the basic counting principle, is a method or guideline for calculating the total number of outcomes when two or more events occur simultaneously. The product of the number of outcomes of each individual event is the total number of outcomes of two or more independent occurrences, according to this concept. For example- if we have to do two things, the number of ways to do one is n and the number of ways to do another is m then we n*m ways to do both the things.
Law of multiplication
Let’s say we have two events, A and B, that are mutually independent, meaning the outcome of one event has no bearing on the outcome of the other. Let E be an event that describes a circumstance in which either event A or event B must happen, i.e. both events A and B must happen. The number of ways in which the event E can happen, or the number of different outcomes of the event E, is then calculated as follows: n(E) = n(A) × n(B). This is known as the law of multiplication.
Law of addition
Let us consider two events, A and B. The number of ways in which event A can occur/the number of possible outcomes of event A is n(A), and similarly, the number of ways in which event B can occur/the number of possible outcomes of event B is n(B). Furthermore, occurrences A and B are mutually exclusive, meaning they have no common result. Let E be an event that describes a circumstance in which either event A OR event B takes place. The number of ways in which the event E can happen, or the number of different outcomes of the event E, is then calculated as follows:
n(E) = n(A) + n(B). This is known as the law of addition.
Conclusion
According to the Fundamental Probability of Counting, if a probability scenario exists in which there are x1, x2, x3… xn entity objects, each with y1, y2, y3… yn options available for each entity, then the number of ways = y1, y2, y3,…, yn. If an event A may occur in m ways and an event B can occur in n ways, the occurrence of both occurrences A and B can occur in m * n ways. This is known as the law of multiplication. If an event A can occur in m different ways and an event B may occur in n different ways, then either of them can occur in m+n different ways. The total ways of choosing are calculated by multiplying the possibilities for each available entity.
