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What is Factorial? - Definition, Examples, Properties, Formula

What Is Factorial? - Definition, Examples, Properties, Formula

Factorials can be easily calculated and have many valuable applications in daily life. For instance, some businesses utilise factorials to examine permutations and combinations for business objectives, such as figuring out how many trucks are required to service each district's establishments.

However, if you hold a position in logistics or work in a sector like finance or software, you can encounter factorial math challenges.

This blog will teach you what a factorial is, how to calculate one, and how to solve issues with factorials.

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What is a Factorial?

The factorial of a whole number, say 'n' can be said as the product of that number with every whole number till its 1.

For example, the factorial of 4 4×3×2×1 is equal to 24.

It is represented using the symbol '!'

Hence, 24 is the value of 4!  

A Brief History,

Do you know that British author Fabian Stedman defined factorial as the equivalent of change ringing in 1677? Meanwhile, the notation n! was introduced by the French mathematician Christian Kramp in 1808.

Numerous mathematical topics, including number theory, algebra, geometry, probability, statistics, graph theory, discrete mathematics, etc., are based on the study of factorials.

How Does Factorials Work?

A number's factorial results from multiplying the integer by each natural number below it. Factorials can be symbolised by the character "!". Thus, n factorial is denoted by n and results from the first n natural numbers!

Consequently, n! or "n factorial" denotes: n! = 1. 2. 3

The product of the first n positive numbers, n, equals n(n-1) (n-2)

i.e., …………………………………n = Product of the first n positive integers = n(n-1)(n-2)

…………………….(3)(2)(1)

Similarly, a factorial calculator works the same way if you run out of time and need an accurate solution for large numbers. You only need to include the numbers you want to find the factorials about. That's all!

Now let's check the examples of factorial so that you can calculate both manually or even with an online factorial calculator-

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Example 1  

4 factorial or 4! is written as:

4! = 4×3×2×1 = 24.

Hence, to find the factorial of a number, multiply the number with the factorial value of the previous number.

Example 2

To know the value of 6!

You need to multiply 120 (the factorial of 5) by 6 and get 720.

Similarly, for 7! multiply 720 (the factorial value of 6) by 7 to get 5040.

What is 0(Zero) Factorisation?

It's common knowledge that the factorial of 0 equals 1. (one). You can write it as 0! = 1.

The above-mentioned notation and definition are justified for a number of reasons. First, the definition produces an extension of the recurrence relation to 0 and allows for a compact expression of many formulae, including the exponential function.

Further, where n = 0, the factorial's definition (n!) includes the product of no integers, making it more broadly analogous to the multiplicative identity.

Besides, there is only one permutation of zero or no objects included in the definition of the zero factorial.

Last but not least, the concept also supports a variety of combinatorial identities.

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Noteworthy Definitions Associated with the Zero Factorial

  • Combinatorics -Mathematical discipline known as combinatorics that focuses on counting.
  • Permutation -Organising a set's members into a linear order or sequence is known as permutation in mathematics.
  • Recurrence relation -In mathematics, a recursively defined sequence or a wide range of values is referred to as a recurrence relation. Recursion is the definition of an object in terms of an object.

Similar Sequences To The Factorial

Numerous sequences that are similar to the factorial exist in mathematics. They consist of:

  • Trigonometric integrals can be made simpler with double factorials.
  • Multiple exclamation points can be used to indicate multi-factorials.
  • Primorials involve multiplying two prime integers whose sum is less than or equal to n.
  • The result of the first n factorials is known as a super-factorial.
  • Hyper-factorials are produced by multiplying several successive numbers between 1 and n.

How to Calculate a Factorial?

You can follow these steps to solve a factorial:

Find the Number

Choose the number for which you are computing the factorial.

For example, an exclamation point and a positive integer both appear in a factorial.

Mathematically speaking, the factorial for the number eight, for instance, would be 8!

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Create the Order/Sequence

You can write out the series of numbers you'll multiply using the factorial formula.

In this case, the number you are calculating the factorial for is eight, along with all the numbers that follow it in decreasing order, down to one.

This is how it would look mathematically:

n! = n(n-1) = 8(8 − 1)(8 − 2)(8 − 3)(8 − 4)(8 − 5)(8 − 6)(8 − 7)

Multiply The Numbers

Once you have the sequence of numbers written down, you can multiply them together.

For example, if you multiply all the numbers in this example, 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, you will get a final answer of 40,320.

Mathematically, it looks like this:

n! = n(n-1)

8(8 − 1)(8 − 2)(8 − 3)(8 − 4)(8 − 5)(8 − 6)(8 − 7) = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

=40,320

Using an online factorial calculator is another option for computing a factorial.

Here is a simple step-by-step process of factoring calculator-

  • There ought to be an "x!" button on the calculator.
  • Press the "x!" button after entering the number you want to calculate the factorial for. In this example, the number eight.
  • You will receive the same result from the calculator, i.e. 40,320.

Here are some example problems that use factorials:

Example 1

Question: Without repeating any letters, how many ways can you arrange the letters in the word "company"?

To solve this problem, count the letters in the word "business," which yields a total of six.

Then, by simplifying with a factorial calculator or calculating manually, determine the factorial of the number six(6).

When you manually solve the problem, it ought to appear as follows:

n! = n(n-1) = 6(6 − 1)(6 − 2)(6 − 3)(6 − 4)(6 − 5) = 6 x 5 x 4 x 3 x 2 x 1 = 720

You now know that there are a total of 720 possible ways to arrange the letters in the word "business" without any repetitions.

Example 2

Question: Find the factorial for the number 15.

While doing it manually is an option, given that 15 is a big number, it can take a while. Using a factoring calculator is simpler.

Follow these steps to calculate through an online factorial calculator -

  • Type the number 15 into your calculator.
  • Press the "x!" button on your calculator.
  • The answer 1,307,674,368,000 should appear on the calculator.

Example 3

Question: What arrangements are possible using red, blue, and green hues?

Since there are three colours in this issue, compute the factorial for that number and then list the various pairings. If you manually solve this issue, the solution would look like this -

n! = n(n-1) = 3(3 − 1)(3 − 2) = 3 x 2 x 1 = 6

The six combinations are:

red, blue, green

red, green, blue

green, blue, red

green, red, blue

blue, red, green

blue, green, red 

Takeaway,

Solving with factorial is as easy as reading numbers if you know them in order. All you need is regular practice, as it's one of the initial processes that will help you with algebra, probability theory, computer science and much more.

So, take this blog as your guide and see how easy it gets to work with factorials!

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