How many different ways can a deck of cards be shuffled? Is there any limit to the number of ways you can arrange your toys? How many different ways are there to arrange the letters in the word candy without having them repeated?

Factorials are the means to knowing the answer to all these above-stated questions.

Now, there are 52! different ways to shuffle a deck of cards.

Similarly, if there are two toys, you can arrange them in two different ways, and if you have three toys, there are six ways you can arrange them.

The word candy has 5 letters, so your answer is 5! = 5 x 4 x 3 x 2 x 1 = 120 ways.

How did we get these numbers for answers? And what does the exclamation mark next to a number mean? How did we make these calculations?

We must first understand factorials in order to understand these.

So, What are Factorials?

A factorial is a function in mathematics with the symbol (!) that multiplies a number (n) by every number that precedes it.1 In simpler terms, factorials are denoted by an exclamation mark and involve repeatedly multiplying all integers up to a predetermined limit.

Therefore, it is likely that you will use a factorial whenever you need to know anything about rearrangements, probabilities, permutations, or combinations. If the order of the components is essential, permutations tell us how many different arrangements are possible. Combinations explain how many different ways there are to select k items from n items when their order is irrelevant.2

Real-life Uses of Factorials

Factorials play a significant role in combinatorics, which is the area of math concerned with combinations of elements from a finite set under specific restrictions, such as those imposed by graph theory, in the real world.

Take a straightforward task as an example: a new piece of code is successful 80 percent of the time. What is the likelihood that all five implementations of the code were successful? How likely is it that at least two trials will be successful? The answer entails computing a few straightforward factorials.

The factorial function can also be used to determine how many different ways there are to select an item from a group of options. Consider the situation where you need to decide what to wear to school each day this week. Consider that although you own nnn articles of clothing, only kk of them have been washed and ironed. How many ways are there to select kkk clothing from a collection of nnn clothing? Sounds like it’s a complex task? It doesn’t have to be because, in situations like these, the factorial function can be pretty helpful.

Factorials are also used in advanced algebra for sequences and series, in calculus for related reasons, in probability, in number theory, and in a number of other disciplines. They are also sophisticated and applied to fields outside of their typical use, like many different aspects of math, and this can be used to create and resolve some highly complex equations.

Factorials can therefore be used in virtually endless situations. Check out BYJU’S FutureSchool Blog if you want to learn more about these subjects and other ways math and its disciplines are used in our daily lives.

References:

1. Factorials: What Are They, How To Calculate Them and Examples | Indeed.com. (n.d.). Retrieved September 1, 2022, from https://www.indeed.com/career-advice/career-development/how-to-calculate-factorial
2. What is factorial used in real life? (n.d.). Retrieved September 1, 2022, from https://iq-faq.com/en/Q%26A/page=a995d91a0e0ecf47eac3b85b1e0406a1
3. Factorials and Their Applications. (n.d.). Retrieved September 1, 2022, from http://www.ltcconline.net/greenl/courses/103b/seqSeries/FACTORI.HTM (general reference)