There are many applications of factorials in other subjects pertaining to mathematics. Factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science (The Factorial Function (Article), 2022). First, let's look at their simple applications. In the paragraph above, a simple scenario was given in which you wanted to find how many ways you could arrange 7 different colored apples. Now, instead of finding out different ways to arrange the
apples, you want to find how many ways you can pick a certain amount of apples from the
group. This can be found using the function: \[n! \over (k!\cdot(n-k)!)\] where k is the number of objects you
want to select and n is the total amount of objects you are selecting from (The Factorial Function (Article), 2022). Say you want to find out how many ways you can select 3 apples from the
group of 7. That would be shown as:
\[7! \over (3! \cdot (7 - 3)!) = 35.\] While this is a relatively simple application of
factorials in math, there are also more complex applications.
A power series is a special type of infinite series representing a mathematical function in the form of an infinite series that either converges or diverges. In terms of factorials, you can use the power series to represent an exponential function. The power series that
represents an exponential function is shown as:
\[e^x=\Sigma_{n=0}^\infty \frac{x^n}{n!} = 1+ \frac{x}{1!}+\frac{x^2}{2!}+ \cdots + \frac{x^n}{n!} \]
(Power Series | Examples of Power Series, n.d.). As you can see, factorials are heavily used in the denominator of the elements in the series. This representation is known as the Maclaurin
series, and has many other uses for expanding functions. Another use of the Maclaurin series is
in the use of the sine function. It is represented as the following:
\[\Sigma_{n=0}^\infty \frac{f^n(0)}{n!}(x-0)^n \]
(Maclaurin
Expansion of Sin(X) | the Infinite Series Module, 2020). Again, notice the repeated use of
factorials in the denominators. Another series that heavily makes use of factorials is the Taylor series, which is a series expansion of a function about a point. The Taylor Series is represented as
the following:
\[\Sigma_{n=0}^\infty \frac{f^n(a)}{n!}(x-a)^n \]
(Weisstein, n.d.-d). Notice that if a=0, you now have the
Maclaurin series. While these are all complex series, it is worth noting that they are all interconnected with each other, have similar constructions, and all use factorials in their computations.
While the simple definition of a factorial is trivial, there are other nontrivial uses of factorials that are rarely used, or taught in any form of mathematics education. There are five extensions of factorials that take the simple definition and expand it to new territories. The multi-factorial is one such extension. A multi-factorial is denoted by any number of exclamation points
and is used to find the product of positive integers
≤ n in steps of how ever many exclamation
points are given(Weisstein, n.d.-a). For example, 3!!! is shown as: \[n!!! = n(n - 3)(n - 6)\cdots\] In the case of two exclamation marks, it is called a double factorial. Double factorials have applications in trigonometry and are used to simplify trigonometric integrals(Team, 2021).
Primorials are another extension of factorials. Primorials are denoted with a "#" instead of a "!" and operate similarly to factorials. While factorials successively multiplying positive integers, the primorial function only multiplies prime numbers(Weisstein, n.d.-b).
The final two extensions of factorials are super-factorials and hyper-factorials. Super-factorials are defined as
the product of the first n factorials. It can be shown as: \[n$=1!\cdot2!\cdot\cdots \cdot n! \] For example, the super-
factorial, \[4$=1!\cdot2!\cdot3!\cdot4!=288\](Weisstein, n.d.-c). Hyper-factorials operate similarly to super-factorials, and
they are the product of the numbers of the form \{1^1 \cdot 2^2 \cdot \cdots \cdot n^n\]. Hyper-factorials can be shown
as the following: \[H(n)= \prod_{i=1}^n i^i \] For example, \[H(3)=1^1\cdot2^2\cdot3^3=108\] (Weisstein, n.d.-a)
As you can see, factorials have a very wide range of applications in mathematics, spanning from power series and Taylor series, to smaller extensions such as double factorials and super-factorials. The significance of these applications allows mathematicians, physicists, computer scientists, and data analysts to solve real world problems that affect the world around you. While the applications that were shown above are methods of solving problems, there are still a plethora of real world applications of factorials.
For most who study math, the interesting part comes with real world applications.
Real World Applications
Knowing how to use factorials in the Taylor Series is neat, but what can it be used for in the real world? The Taylor series can be used by electrical engineers to analyze power flow of electrical power systems(Weisstein, n.d.-d). The Maclaurin series is used by scientists and many different fields of engineering as well(Maclaurin Expansion of Sin(X) | the Infinite Series Module, 2020). The five extensions of factorials can aid in the process of simplifying calculations by those scientists, mathematicians, and engineers. All of these methods of mathematics have their own unique ability to help many different professions better the world. For those who aren't scientists or engineers, there are also unique applications in the real world that allow us to stretch our thinking on such a seemingly simple concept.
A deck of cards consists of 52 unique cards. Imagine you are at a magic show, and the magician is doing a card trick. He shuffles the cards and lays the whole deck in front of you. You look the entire deck over and notice that all the cards are accounted for, and in random order. The magician cannot see the order of the cards. Ask yourself if this is a unique shuffle of cards? The answer is probably yes. In fact, there is a very strong chance that the deck of cards you just looked at is unique, and the first time that
arrangement has ever been randomly shuffled in the history of human civilization. This seems impractical, but let's observe why using factorials.
As stated earlier, factorials can be used to calculate the number of ways to arrange n objects. In this case, we are concerned with how many ways to arrange a deck of 52 cards. Thus, the calculation that has to be done is 52!. Before you begin to calculate this, it should be noted that it will take you an incredibly long time. As you can imagine, 52! is incredibly large. When you calculate the final answer you should come up with the following number:
\[80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000\]
This number is so large, it is difficult for the human mind to comprehend. For the sake of comprehension, it is worth examining exactly how large this number is by playing a game. To help aid in the visualization of this game, i've linked a video from youtube at the bottom of this page.
Game
Start a timer that will count down the number of seconds from 52! to 0. Let's see what can happen in that span of time. Begin by picking your favorite spot on the equator. You're going to walk around the world along the equator, but take a very slow pace of one step every billion years. If you feel inclined, it may be a good idea to bring a deck of cards to play a couple billion games of solitaire. Continue along the equator until you complete your trip around the world.
Once complete, remove one drop of water from the Pacific Ocean. Now do the same thing again: walk around the world at one billion years per step, removing one drop of water from the Pacific Ocean each time you circle the globe. Continue until the ocean is empty. It is worth noting that the equatorial circumference of the Earth is 40,075,017 meters, and the Pacific Ocean contains
707.6 million cubic kilometers of water(52 Factorial, n.d.).
Once the Pacific Ocean is completely empty, take one sheet of paper and place it flat on the ground. Now, fill the Pacific Ocean back up and start the entire process all over again, adding
a sheet of paper to the stack each time you've emptied the ocean. Do this until the stack of paper reaches from the Earth to the Sun. The distance from the Earth to the Sun is described as 1 Astronomical Unit, and is defined as 149,597,870.691 kilometers(52 Factorial, n.d.).
Once the paper reaches the Sun, take a glance at the timer, you will see that the three leftmost digits haven't even changed. So, take the stack of papers down and do it all over again. In fact, do it 1000 more times; taking one step every billion years, emptying the Pacific Ocean one drop at a time once you circle to globe, lay one sheet of paper once the ocean is empty, and repeat until the stack of papers reaches the Sun. You may think you’re close, but you’re just about a third of the way done(52 Factorial, n.d.).
Of course, this game is incredibly unrealistic, and the Earth would be swallowed up by the expansion of our dying Sun by the time you take your fifth step, but it relays an important message about factorials. When studying such a simple concept, it is important to open your mind to the vast possibilities it can portray.
Now, going back to the magic show. After the magician shows you the cards which he cannot see, he asks you to reshuffle them. Once you've shuffled the cards, the magician then takes the deck, and randomly selects each card from the deck and lays them out in front of you. By the time he lays the last card, you notice that he perfectly replicated the original order of cards.
What are the odds of this happening in real life? The odds of selecting the first card to be correct is relatively simple; it's
\[\frac{1}{52}\]
The odds of selecting the first two correct are:
\[\frac{1}{52}\cdot\frac{1}{51}=\frac{1}{2652}\]
The odds of selecting the first three correct are:
\[\frac{1}{52}\cdot\frac{1}{51}\cdot\frac{1}{50}=\frac{1}{132600}\]
(Teixeira, 2017). As you can see, factorials will have a
major use in this solution. Part of the beauty of factorials is that they allow you to concisely write out problems that would be otherwise difficult. The odds of the magician randomly selecting the
entire deck of cards in order is:
\[\frac{1}{52!}=\frac{1}{8.07\cdot10^{67}} \]
This is an incredibly low chance.
Factorials have many uses, some simple, others more advanced. Their history has been examined for hundreds of years by multiple mathematicians like Bernoulli, Goldbach, Euler, and Kramp. The applications in the mathematics world allow professionals in other disciplinnes to complete complex solutions that benefit society. Their extensions allow mathematicians to compute solutions in less time. More so than anything, they use a seemingly simple idea, and extend it to every part of the real and mathematical world.