Principles Of Counting

An off-branch of mathematics, combinatorics focuses solely on the study of countable and finite discrete objects. Combinatorics can also be used in statistical physics, computer science, and optimisation. While it is generally possible to count the number of outputs that may come out of an event by simply glancing at each possible outcome, this method is ineffective when dealing with a large number of outcomes. This is where the principle of counting is used.

Consider a situation where you are given choice A with three options, A1, A2 , and A3 , and choice B with only two options, B1 and B2. If you have to first make a selection from option A and then from option B, the total number of outcomes possible can be 3 x 2 = 6. Thus, the total outcomes are A1 B2, A2 B1, A2 B2, A3 B1, A3 B2.

You can use the basic principles of counting by creating a decision tree to reach these six outcomes. The decision tree is basically a graph that will model the possible outcomes at every stage of your experiment. To create a decision tree, you will first need to determine the number of decisions you will be making. In the example above, only two possible decisions can be made. 

1. You can choose an outcome from option A. 
2. You can choose an outcome from option B. 

In the next step, you will be required to think of possibilities that may arise from the first choice, which, in our case, is three, and the number of possibilities that may result from the second choice, which, in our case, is two. Thus, according to the fundamental counting principle, you need to multiply those two numbers to get the final total number of outcomes that can result from the situation. 

Formulas of Permutation and Combination

Another type of situation of counting can arise when you have been provided with a certain number of objects, and you have been asked to select some or all options, and need to know the number of ways you can do this. For instance, imagine a situation where a teacher has a class with 30 students divided into groups of five each to do a presentation. Now the teacher wants to know the number of ways in which it can happen. This situation can be solved with combination and permutation formulas. However, the sole difference between permutation and combination lies in whether or not the order through which you are selecting objects matters. Thus the importance of the basic principles of counting is undeniable. 

If the teacher is trying to choose a group to create a presentation, it will be a problem of combination as the order will not matter here.

If the teacher needs to choose the first, second, and third place holders in the science exhibition competition, it will be a problem of permutation as the order will not matter here.

The factorial symbol “!” is used to multiply each natural number, including that whole number itself. Thus, 5! will be 5, 4, 3, 2, 1. This factorial symbol is commonly used with permutation and combination formulas. 

Several formulas can be used with the concept of permutation and combination. 

However, there are two main formulas:

The formula of permutation: Permutation refers to the selection of “r” objects selected from a set of n objects that have no repetition and where the order must matter. 

nPr = ( n ! ) / (n – r ) !

The formula of combination: Combination is when the choice of “r” objects from a group of “n” objects has no repetition, in which case, the order does not matter. 

nCr = nPr / r ! = n ! / r ! (n – r) !

Solved Problems Based on Permutation And Combination Formulas

If you are going on a trip with four friends in a vehicle that can only fit five persons, what is the total number of different ways that every person can sit in the vehicle if you are the one driving?

Solution. The fundamental counting principle is very useful in such situations. As the same person will consistently occupy the driver’s seat, only four options can occur on the passenger’s seat. Once that person has been seated in the passenger seat, only three other options are left for the next available seat. This same process will continue until only one person is left with only one seat.

Therefore, 1. 4. 3. 2. 1. = 24.

The total number of different ways each person can sit in the vehicle if the driver remains the same is 24.

Conclusion

Combinatorics is the study of countable and finite discrete objects. Combinatorics can also be used in statistical physics, computer science, and optimisation. The permutation is the selection of “r” objects that have been selected from a set of n objects which has no repetition and where the order must matter. The combination is when the choice of “r” objects from a group of “n” objects has no repetition, in which case, the order does not matter.