FUNDAMENTAL PRINCIPLE OF COUNTING

EXPLANATION

The Fundamental Counting Principle formula is a simple, intuitive principle in mathematics, that we observe in our real lives rather often. It is basically a method to find out the number of possible outcomes, or all the possible ways of doing something with a given number of events. Let us try to understand this with some relatable examples:

Imagine you have a family function coming up and you are trying to put together the perfect outfit for the day. You have 5 shirts, 4 pairs of trousers, and 6 different ties that you feel will be appropriate for the occasion. Now, you can match each shirt with four different pairs of trousers, and for each shirt-trouser combination, you can try 6 different ties. That is a whole lot of options to choose from. So, to find out exactly how many possible unique outfits you would have, you simply need to multiply 5 x 4 x 6. That is what the Counting Principle Formula is all about. 

If you think about it, the Counting Principle formula makes finding out all the possible outcomes when there are two or more events a lot easier. (Here, having different shirts, trousers and ties are the different events.) Otherwise, to count the same thing, you would have to use addition: for each shirt, you can wear 4 trousers: therefore, the total being 4+4+4+4+4, and for each shirt/trouser combination, would have 6 options for ties. That is, (4+4+4+4+4) + (4+4+4+4+4) + (4+4+4+4+4) + (4+4+4+4+4) + (4+4+4+4+4) + (4+4+4+4+4), which is a very tedious calculation. So, instead of doing all of that work, you only need to multiply the different events together. It is important to note that for the Counting Principle to apply, these events are supposed to be independent of each other. That means the choice of shirt or trousers or ties should not depend on each other in any way. If you have a pink shirt that would only go with a certain black pair of trousers, then the principle would not work. 

Let us see another example. Imagine you go to your favorite ice cream place that allows you your choice of cone, flavor, and toppings. You could either take your ice cream in a cup, or a cone, or a waffle base. The 5 flavors  to choose from are chocolate, vanilla, butterscotch, mango, and strawberry. The toppings are almonds, gems, sprinkles, or choco chips. Now with all these different choices, which are known as events in mathematics, how many different types of ice creams can you have? The answer is 3 x 5 x 4, which is 60. Again, a lot of options to choose from!

The Fundamental Counting Principle formula is very useful in real life, as it predicts the number of ways that the given events can occur simultaneously. This is also what forms the basis of other mathematical concepts like probability and permutation/combination.

Say, for example, you went on a holiday and your suitcase had a three-digit numeric lock. If by chance you forgot the combination of the lock, how many possible combinations would you have to try to find the code? For a three-digit lock, where each digit can be anything between 0 and 9, the total number of possible combinations will be 10 x 10 x 10, which is 1000. Thus, this principle will be very important in higher mathematical concepts, as well as real-life decisions and scenarios. It also shows how you should not forget the passcode of your lock so as to avoid trying a thousand different combinations to open your suitcase. 

REAL-LIFE APPLICATIONS

The Counting Principle formula has several real-world uses. In its more complex form, it is used in almost every branch of mathematics, in computer science for sorting algorithms, artificial intelligence, molecular biology, quantum physics for describing the states of particles, in biology for describing RNA sequences, and even linguistics, word processing, and databases.