Understanding Factorials

Have you ever wondered about the number of ways your books can be arranged in your bag or on a shelf? Initially, it may appear simple to count, but as the number of items grows, the number of possible arrangements increases exponentially, so much so that it is almost impossible to list each one. This is where factorials come into the picture. A powerful mathematical tool for calculating the number of ways a set of things or numbers can be arranged. Here, we will understand this tool along with some special cases and applications in mathematics or other fields of study. 

1.0Factorial Meaning 

In mathematics, the factorial of a non-negative integer n is the product of all positive integers from 1 to n. It is a basic concept applied in counting problems, such as to find the number of ways a set of objects can be arranged. This concept is a simple topic yet holds much importance in many areas of mathematics, which we will be discussing in the coming sections. 

2.0Factorial Notation 

Factorials in mathematics are denoted with any number, say n, with an exclamation mark “!” placed after the number, like this: 

$n!=n\times (n-1)\times (n-2)\times \dots \times 2\times 1$

Here: 

• The symbol ! is read as “factorial”.
• n! Holds true for all non-negative integers.

3.0Properties of Factorials 

Factorials possess several dynamic properties, which ultimately help in solving problems related to factorials with ease: 

1. Recursive Relationship: A factorial shows a recursive relation with its subsequent terms, meaning it can be defined in terms of the preceding factorial, like this: 

n!=n(n-1)!

1. Rapid Growth: As mentioned earlier, values of factorials grow rapidly. For example, the value of: 

10! = 3,628,800

20! = 2.43×10 

3. Non-Negative Integers Only: Only non-negative integers have factorials in standard mathematics. Factorials are not found in negative numbers in elementary mathematics.
4. Multiplicative Identity: As n! deals with multiplication, it has the common multiplication laws like associativity and commutativity.

4.0Factorial Numbers: Special Cases 

Although there are countless important factorial numbers, some of them are special to understand for solving questions related to the factorials, such as: 

0 Factorial: 

In simple algebra, multiplying any number by 0 becomes 0. However, in a factorial system, multiplying any number by 0 factorial does not give 0 due to its value being equal to 1. Which simply means: 

0! = 1

1 Factorial

It is important to understand that 1 factorial always gives 1 only, as the number can not be decreased further, because factorials are used only for “non-negative integers”. Which simply means: 

1! = 1

5 Factorial

When solving questions related to factorials, either arrangements or selections, 5 factorial is an important entity, which equals: 

5! = 200

100 Factorial

100 factorial is a very large digit with 158 total digits to be multiplied with one another. Which can be a scorching and almost impractical task. Hence, advanced tools and codes are used to calculate such big factorials. 

5.0Calculating Factorial 

To find a factorial is to determine the product of all positive whole numbers from 1 to a certain number. The factorial of a number n, denoted as n!, is found by using the formula stated above with these easy steps:

• Begin with the given number (n).
• Multiply it by each smaller positive number down to 1.
• Record or simplify the answer.

6.0Where is Factorial Used?

The fundamentals of factorial forms a base for many important topics of mathematics and other fields of study. Here is a list of the main areas of factorials, where they are used: 

1. Permutations: Although permutations are a much more advanced topic than factorials, it is helpful to know how they work and how factorials play an important role in solving these questions. Permutations are involved when things need to be arranged in order. It has a specific formula which has a factorial as its main component, which is: 

$P(n,r)=\frac{n!}{(n-r)!}$

• n is the total number of items in a set of numbers or things. 
• r is the number of items to be selected from the set of numbers. 

2. Combinations: In combinatorics, which is that part of mathematics concerned with counting, factorials are applied to determine the number of ways objects can be chosen from a set, particularly when it doesn't matter where they are placed.

$C(n,r)=\frac{n!}{r!(n-r)!}$

3. Probability: Factorials also help in calculating the probability of an event, to find out the total number of possible outcomes and favourable outcomes when determining the. 
4. Binomial Theorem: In algebra, the binomial expansion of expressions involves binomial coefficients, which are determined by factorials. Every term in the expansion is determined by:

$(x+a{)}^n={\sum }_{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^k{a}^{n-k}$

Here, 

• n is a non-negative integer,
• $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$is the binomial coefficient, calculated using factorials: 

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{r!(n-r)!}$

5. Algebra and Series Expansions: For infinite series of algebra, in advanced mathematics, the factorial formula is used to solve the function, such as for the exponential function, the formula used is: 

$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$

7.0Solved Example of Factorials 

Problem 1: Evaluate 7! − 4!

Solution: According to the question, 

$\begin{array}{rl}& 7!=7\times 6\times 5\times 4\times 3\times 2\times 1=5040\\ & 4!=4\times 3\times 2\times 1=24\\ & 7!-4!=5040-24=5016\end{array}$

Problem 2: Evaluate $\frac{6!}{3!}$

Solution: According to the question:

$\begin{array}{rl}& \frac{6!}{3!}=\frac{6\times 5\times 4\times 3!}{3!}\\ & \frac{6!}{3!}=6\times 5\times 4=120\end{array}$

Problem 3: How many ways can 5 different books be arranged on a shelf?

Solution: According to the question: 

The number of books to be arranged = 5

The number of ways these books can be arranged = 5! = 120