Profile
Settings
Refer your friends
Sign out
Counting Theorems are ways to calculate and arrive at a solution when both the variables to be added or multiplied are not fixed. For instance, you want to eat dinner at a restaurant. So, you get into your car to arrive at any restaurant. Here, the two variables, A(the restaurant) and B(the route), are not fixed. Therefore, A and B are also interdependent. There can be multiple ways to solve such a case. These ways are known as permutations and combinations, and the calculation system is known as the Counting Theorem.
A permutation is how a set of objects can be arranged or how an incident can occur. Let’s take an example of six friends named Jaya, Alka, Mira, Rima, Piya, and Seema who want to sit in a row at the cinema. There are six seats available.
To represent this calculation in a formula, we can say that there are 6×5×4×3×2×1=720 ways of arranging six friends on six seats. You will notice that both the number of friends and the number of seats are finite numbers here. This is also called the rule of multiplication.
A combination is a selection of items such that the arrangement of the items does not matter. For example, we arrange the letters A, B, and C. In a permutation, the account ABC and ACB are different. While, in combination, the arrangements ABC and ACB are the same because the sequence does not matter.
The counting theorem is used frequently in our daily lives while setting passwords, making timetables, selecting combinations of clothes, selecting menus, etc. It is almost overlooked when there aren’t large numbers involved. It helps predict large numbers. For example, companies dealing with massive customer bases or products. It is widely used in data sciences for samples and surveys to interpret conclusions.
Introduction
Counting Theorems are ways to calculate and arrive at a solution when both the variables to be added or multiplied are not fixed. For instance, you want to eat dinner at a restaurant. So, you get into your car to arrive at any restaurant. Here, the two variables, A(the restaurant) and B(the route), are not fixed. Therefore, A and B are also interdependent. There can be multiple ways to solve such a case. These ways are known as permutations and combinations, and the calculation system is known as the Counting Theorem.
The counting theorem simplifies mathematical problems by substitution addition or multiplication. It is only used in situations where there are no fixed variables, but the range of the variables is known. For example, A is a variable event that can happen in ‘n’ number of ways. Similarly, B is another variable event that can occur in a ‘p’ number of ways. Thus, to find out the number of ways both A and B can be done, we will use the counting theorem. The solution will also be a range of values instead of a fixed number. This is the basic underlying principle of permutation and combination.
Permutations
A permutation is how a set of objects can be arranged or how an incident can occur. Let’s take an example of six friends named Jaya, Alka, Mira, Rima, Piya, and Seema who want to sit in a row at the cinema. There are six seats available. So, in how many ways can they be seated? Understanding this practically;
- For the first seat, there is a choice of 6 friends.
- Once the first person is sitting, we select the remaining five friends for the second seat.
- After filling two chairs, we have a choice of the remaining four friends for the third seat.
- After seating the third person, we have a choice of any of the remaining three friends for the fourth seat.
- After seating the fourth person, we choose the remaining two friends for the fifth seat.
- Lastly, we select only 1 of the remaining friends for the sixth seat after seating the fifth person.
To represent this calculation in a formula, we can say that there are 6×5×4×3×2×1=720 ways of arranging six friends on six seats. You will notice that both the number of friends and the number of seats are finite numbers here. This is also called the rule of multiplication.
Combinations
A combination is a selection of items such that the arrangement of the items does not matter. For example, we arrange the letters A, B, and C. In a permutation, the performance ABC and ACB are different. While, in combination, the arrangements ABC and ACB are the same because the sequence does not matter. What matters is the selection. So if two items are to be selected, then AB, BC, CA, etc., all apply without the arrangement mattering.
Conclusion
The counting theorem is used frequently in our daily lives while setting passwords, making timetables, selecting combinations of clothes, selecting menus, etc. It is almost overlooked when there aren’t large numbers involved. It helps to predict large numbers. For example, companies dealing with massive customer bases or products. It is widely used in data sciences for samples and surveys to interpret conclusions. They form an essential part of computer programming as well. In mathematics, it is used in concepts such as partitions. It can be understood that the uses of the counting theorem are infinite. They are used in the world of technology, which is one of the most important applications of mathematics.
