The factorial of an integer is very easy to calculate and involves multiplying a positive value starting from with all the positive values lesser than it. Factorials have been applied in the business world in different ways. It is used for determining permutations, probabilities, and combinations. For example, logistics companies are one of the biggest beneficiaries of Factorials as they use them to determine how to distribute goods to different destinations.
Factorials are very easy to calculate and do not require any complex math. It is totally about multiplications. In this article, you will find out how to calculate factorial effectively.
Factorial Formula
The factorial of a positive integer (n) is the product of the integer n with all of the values lesser than it. The factorial is denoted by !.
The mathematical formula of the Factorial of n is
n! = n x (n-1) x (n-2) x …x 2 x 1
n! = n x (n-1)!
How to Calculate Factorial
These are steps to calculate a factorial.
- The value must be a positive integer starting from 1
- Multiply the integer with all of the integers lesser than it in a descending order
- The factorial of 0 is 1.
What Is Factorial
Generally, the factorial of an integer is straightforward, and all a mathematician needs to do is simply multiply downwards. Several examples will be considered to give a clear understanding of calculating an integer.
Example 1
Find the factorial of 4
solution
n! = n(n-1)!
4! = 4 x 3 x 2 x 1 = 24
Example 2
Calculate 0!
Solution
By the definition of factorial, 0! = 1
There is a foundational reason for this. However, the bottom line is that looking for an empty integer requires you to do nothing. The fact that you are doing nothing is considered an event or action because you made a (1) decision to do nothing. Hence 0! = 1.
Example 3
Calculate the following expression 5!4!3!
Solution
Find each factorial in the expression
5! = 5 x 4 x 3 x 2 x 1 = 120
4! = 4 x 3 x 2 x 1 = 24
3! = 3 x 2 x 1 = 6
Therefore, the expression will be
How to Calculate Complex Factorials
Clearly, from the first three examples, computing factorial is pretty much straightforward. However, all the numbers calculated were small values that could easily be figured out. There are cases when you may need to calculate the factorials for more complex numbers like 45! At that point, trying to calculate the factorial manually may leave you with complex calculation patterns and a high chance of getting a wrong answer.
Fortunately, there is a way you can break down extremely large values and still get the right answer. However, it is important that you follow the rules of the factorial to get it right.
1st Rule: The factorial of a value is not equal to the factorials of its factors except when the value is multiplied by 1. For example, 18! 9! X 2!
2nd rule: break down the digits into smaller factorials and calculate.
The two rules will be expressed in example 4 and example 5, respectively.
Example 4
Show with the factorial formula that 18! is not equal to the factorials of its factors when any of the multiple is not 1.
Solution
Determine all the factors of 18.
The factors of 18 are;
18 and 1
9 and 2
6 and 3
2, 3, and 3
The next step is to calculate 18! and that of the various factors to show they are not equal
The formula for factorial = n(n-1)!
For 18!
18! = 18 x 17 x 16 x 15 x 14 x 13 x 12 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 6402373705728000
For 9 and 2
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880
2! = 2 x 1 = 2
9! X 2! = 362,880 x 2 = 725760
For 6 and 3
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
3! = 3 x 2 x 1 = 6
Therefore, 6! X 3! = 720 x 6 = 4320
For 3, 2 and 3
18 = 3 x 2 x 3
3! x 2! x 3!
The next step is to calculate each factorial and add it together
3! = 3 x 2 x 1 = 6
2! = 2 x 1 = 2
Therefore;
3! x 2! x 3! = 6 x 2 x 6 = 72
Clearly from the varying answers 18! 9! X 2! 6! X 3! 3! x 2! x 3! and as such, the factorial of 18 is not equal to the factorials of its factors.
Example 5
Calculate the factorial of 55
The factorial of 55 is an amazingly long number, but the focus in this example is not really about the answer. Rather, the focus is on how to break down the computation sequence conveniently.
55! = 55 X 54 X 53 X 52 X …..X 3 X 2 X 1 = 12,696,403,353,658,275,925,965,100,847,566,516,959,580,321,051,449,436,762,275,840,000,000,000,000
55! =55 X 54 X 53 X 52 X 51 50!
50! = 50 X 49 X 48 X 47 X 46 X 45 X 44 X 43 X 42 X 41 X 40!
40! = 40 X 39 X 38 X 37 X 36 X 35 X 34 X 33 X 32 X 32 X 30!
30! = 30 X 29 X 28 X 27 X 26 X 25 X24 X 23 X 22 X 21 X 20!
20! = 20 X 19 X 18 X17 X 16 X 15 X 14 X13 X 12 X 11 X 10!
Now 10! = 10 X 9 X 8 X 7 X 6 X 4 X 3 X2 X1 = 3, 628, 800
Since 10! Has been identified, the calculation only needs to go upward. The good thing about this systemic arrangement is that it makes the calculation cleaner, with less probability of making a mistake.
Nevertheless, it is important to note that most math tests will only require students to factor values no more than 10. Any more will be too time exhausting and may overwhelm most calculators.
Conclusion
This study has discussed how to calculate factorials. From all that was done, it is obvious that finding the factorial of a positive integer is one of the easiest mathematical processes. All you have to do is follow the descending order of multiplication to get the right answer.
As stated in this article, there is barely any need for you to caudate the factorials of large numbers (numbers going past 20) as they are usually vast and extremely large and will demand too much time. However, easier 1 digit numbers are straightforward, and you should not have a problem calculating them.
