On a bright sunny day, you go shopping. You select a nice pair of jeans and decide to pay using your credit card. But you suddenly realize that you are not able to recollect your pin. What a tragedy!! Now all you can think of is to list all the possible combinations to figure out your pin. How many possible combinations can you make? The answer to this question is difficult if we keep listing each possible combination and counting. In situations like these, the fundamental principle of counting or the multiplication principle comes to our rescue. Let us see what the fundamental principle of counting is all about.

## Fundamental Counting Principle Notes

Suppose you have 2 pairs of shoes and 3 pairs of socks. In how many ways can you wear them? Now, the possible ways of choosing a pair of shoes are 2, since 2 pairs of shoes are available. With any pair of shoes, any of the 3 pairs of socks can be worn at a time. Therefore, for each pair of shoes, there are 3 choices of socks. Similarly, with 2 pairs of shoes, there are 6 choices of socks available since 2 × 3 = 6. This can be understood more clearly with the help of the following figure. Let A_{1} and A_{2} represent the 2pairs of shoes and B_{1}, B_{2}, B_{3} represent the 3 pairs of socks.

In the problem stated above, we use the fundamental principle of counting to get the result. The multiplication principle states that if an event A can occur in x different ways and another event B can occur in y different ways, then there are x × y ways of occurrence of both the events simultaneously.

This principle can be used to predict the number of ways of occurrence of any number of finite events. For example, if there are 4 events which can occur in p, q, r and s ways, then there are p × q × r × s ways in which these events can occur simultaneously.

Well, the answer to the initial problem statement must be quite clear to you by now. The credit card pin would involve a sequence of 4 digits and each digit can vary between 0 – 9. Thus, the number of ways of occurrence of each digit is 10. Since it’s a 4-digit pin, the number of possible combinations is 10 × 10 × 10 × 10 = 10000. There are 10000 combinations possible, out of which 1 is correct. Well, good luck trying to figure that out. This explains to us the fundamental principle of counting, which lays the foundation for permutations and combinations.

**Dependent or Independent?**

The fundamental principle of counting only works when the choice that is to be made are independent of each other.

If one possibility depends on another, then a simple multiplication does not work.

## Frequently Asked Questions – FAQs

### What is the fundamental principle of counting?

### Suppose you have 3 pairs of shoes and 4 pairs of socks. In how many ways can you wear them?

So, we can wear it 3 × 4 ways

I.e., 12 ways

### What is the meaning of the fundamental principle of counting (FPC)?

### What are the concepts for counting?

Multiplication Principle

### If you go outside to buy sweets and suppose a bakery has a selection of 15 different cupcakes, 20 different doughnuts, and 13 different muffins. If you are to select a tasty treat, how many different choices of sweets can you choose from?

We have to choose from either a cupcake or doughnut or muffin,

So, we have 15+20+13 = 48 treats to choose from.

To learn more about the fundamental principle of counting, permutation, and combination, download BYJU’s- The Learning App.

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