Fundamental Principle of Counting

Introduction

The fundamental principle of counting is one of the crucial topics in mathematics that is widely used while solving problems, especially in probability. It is used in problems where we need to find a number of ways in which an event can occur. 

Fundamental principles of counting contain various formula types that even include constant, trigonometric functions, polynomials, permutations, combinations, and probability. There are various mathematical problems and their solutions based on this chapter. Using this principle, you will be able to understand the logic behind many probabilities and permutation problems. 

The fundamental counting principle expresses that if there are ‘a’ ways to do one thing and ‘b’ ways to do some other thing, then there are a × b ways to do both things. 

Let’s understand this using an example:

Assume that you have a 6-sided die and you roll it. You have a deck of cards, and you draw a card from it. Now, there are 6 possible outcomes on the die, and 52 possible outcomes from the deck of cards. So, calculating the total number of possible outcomes of the whole experiment would give us 6 × 52 = 312.

Fundamental Counting Principle Formula:

The principal formula for the fundamental counting principle is the same as its explanation tells. It means, if we have ‘x’ ways/options to do the first task and ‘y’ ways to do the second task, then the total number of ways we can do the first task and second task together is x * y.

Applications of fundamental counting principle:

Now, let’s understand some applications of their principle where it can be used. 

Let us assume that there are 5 girls to be seated on a bench in a row. Let us take them as p, q, r, s, and t. Now, how will we find the number of ways in which they can be seated on the bench? Now let’s take girl p = event p, girl q = event q, and so on. For event p, we have a total of 5 options. For option q, we have a total of 4 options. In the same way, for event r, we have 3 ways. For event s, we have 2 ways and for event t, we have only 1 way.  

In this assumption, to find the total number of ways to do some tasks, we use the counting principle. Using this, we multiply the possibilities of each event. Here, the total number of possible outcomes is 5×4×3×2×1=120. This is also called a permutation, and it is one of the applications of the counting principle.  

Permutation:

The permutation of ‘n’ number of different elements taken ‘r’ at a time such that r ≤ n, then the order of r elements can be shown by P(n,r) = n!n-r!

where n! = n × (n−1) × (n−2)…..×1

Same as permutation, the combination is also an application of the principle of counting. 

Combination:

The combination is the process of choosing elements from a set without considering the order of its arrangement.  For example, let’s assume we have a class of four boys taken as P, Q, R, and S. We want to make a team of three boys only. Now, how many different teams can we make using all the boys? Here, in this case, note that the team {P, Q, R} is the same as the team {Q, P, R} or {R, P, Q}. we meant to say that order doesn’t matter here in this example. Such types of arrangements are known as combinations. To solve these types of problems, we can derive a formula to find the total number of combinations.

In the above example, if we are taking that order doesn’t matter. Taking this, let’s consider the example we used in permutations. By the counting principle, we have a total of 4×3×2×1 possible arrangements. Now for any given order, let’s say {P, Q, R}, there are many ways such as {Q, R, P} or {Q, P, R} that result in the same combination. So, here we are required to find the total number of possibilities of any arrangement that will give the same team.  

Since we only have three slots and three members (P, Q, R), the first slot can be occupied in three different ways, the second alit can be occupied in 2 different ways, and the third slot can be occupied in only 1 way. Now using the counting principle, we will have 3×2×1 total possibilities that lead to the same combination. So, the total number of unique combinations is

4×3×2×1×3×2×1

On a general basis, if we have n elements and we choose r elements to make a combination, then the total number of combinations is given by C(n,r) and is given as C(n,r)=n×(n−1)×(n−2)×⋯(n−r)/r×(r−1)×(r−2)×⋯1

Now, multiplying both denominator and denominator by (n−r)! and by the factorial rule, we can derive that:

nCr=n!r!(n-r)!

In this way, you can solve many problems using the counting principle.

Conclusion

This was all about the counting principle which is used to solve many problems based on permutations and combinations.