Fundamental Principle of Counting

There is a particular definition for the fundamental principle of counting and needs to be understood with examples for more clarity. The total number of outcomes of two or more events taking place independently can be derived as the product of the number of outcomes resulting from each individual event. A real-life example can be given in this context. Suppose, a child has to choose among 6 different flavours of ice cream and 3 different types of cones at this case, the child can have access to 6 X 3 = 18 different choices concerning ice creams. 

Overview of the principle of counting

The principle of counting plays a significant role in our daily life. It helps us to find the number of ways an event can take palace or the possibility of outcomes that can be generated from multiple events. It can be used while choosing an outfit, such as the resultant outfit as an outcome of the combination of different shirts and pants. Moreover, the combination of drinks and meals can be determined by determining the outcomes by applying the principle of counting. 

In case an event takes place in m different ways, and another event takes place after that in n number of ways, the number of occurrence of events can be stated as m*n*p. However, it can be observed that counting takes place with permutation and combination. There is a link between the counting principle and permutation. In other words, suppose there are t ways of performing a job, and q ways to do another job, then how both the jobs can be done are p x q ways.  The principle of counting serves as a foundation to find solutions so permutation and combination questions. 

Principles of permutation and combination

A permutation is nothing but the way of arranging and choosing objects with the help of some principles or formulas. On the other hand, the combination may be defined as how K objects from a sum of n things can be chosen. The order of the arrangement of the formula does not matter as the formula states “nCk = (n,k) = n! k! (n-k)!”. There is a subtle difference between the permutation and the combination as permutation is the way of arranging objects whereas in the case of combination it is just about selecting several subjects from a group or collection. It needs to be considered that the arrangement or order of the objects does not matter. Permutation and combination derive their basis from the fundamental principle of counting. There are certain rules associated with the principle of counting such as addition, multiplication, etc. Permutation and combination help to solve the issues of probability or the outcome of events. The basic concepts of probability need to be understood to grasp the idea of permutation and combination principle.

There are many ways permutation and combination can help in testing hypotheses and probability distributions. In the case of permutation, the order is not important; therefore there are some cases in which it is not applicable. Suppose a PIN of an ATM card is given as 9 10 8, then it has to be in the same order, therefore the principles of permutation are not applicable. Details play an important role in permutation as and the order is important enough to be maintained to some extent. 

The combination is quite different from the permutation as details hardly matter in this case, as the arrangement needs to take place in order. It needs to understand the proper use of permutation and combination. In the case of lists, permutations are used, whereas combinations are mostly used for groups where the order has no place. 

Permutation and combination questions are a must in IIT JEE Mains and JEE Advanced. However, Permutation and combination questions come in higher secondary mathematics to help the students deal with questions or situations that involve a large number of sets.  Permutation enables the performer to select subsets considering the order of selection. In the case of combination, subsets are made by selecting objects from sets. In the case of combination, the ordering details need to be eliminated unlike the concept of permutation..

Conclusion

Permutation and combination principle is a significant chapter that helps the students to excel in higher mathematics. It is an important chapter as permutation combination questions are a must in all joint entrance examinations. The concepts of Permutation and combination should be made clear to deal with vast sets of numbers and data or determine the outcome of events.