Factorial Notation: Introduction 

Factorial notation for every natural number will be as n!. According to the factorial notation definition, n! will be equal to the product of n and all its positive preceding numbers. For example, if we have 4!, then this means it shall be written as 4×3×2×1. A French mathematician Christian Kramp in the early 1800, introduced this method as n!. Therefore the factorial of 4 shall be 24. The factorial has great importance in permutation and combination. A permutation is a mathematical method that tells us the total number of ways in which we will be able to arrange elements in a certain combination. Whereas the combination tells us about the total number by which we will be able to select elements from the available set.

Factorial Notation Examples

Factorial notation acts as the base for a variety of mathematical operations. For example, algebra, geometry, statistics, discrete mathematics, probability, permutation and combination. It is a simple method of calculation just by multiplying the preceding and equivalent numbers of given ‘n’. Let’s learn some factorials of natural numbers.

• 0! = 1
• 1! = 1
• 2! = 2 x 1 = 2
• 3! = 3 x  2 x 1 = 6
• 4! = 4 x 3 x  2 x 1 = 24
• 5! = 5 x 4 x 3x  2 x 1 = 120
• 6! = 6 x 5 x4 x3 x 2 x 1= 720
• 7! = 7 x 6 x5 x 4 x3 x 2 x 1= 5040
• 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1= 40320
• 9! = 9 x 8 x7 x6 x 5 x 4 x 3 x 2 x 1= 362880
• 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1= 3628800

Factorial Notation Formula

As already discussed, the factorial of a number is the function of the product of all its preceding numbers. Let’s assume ‘n’ as the factorial under calculation; therefore, the factorial of n! will be equal to the product of n with all its preceding numbers. By this statement, we can write factorial notation formula of n as:

n!=n×(n−1)! 

The ‘n’ here shall be 1 x 2 x 3 x ……n. Therefore, 6! shall be equal to 6×5! (according to the formula). This formula helps us in getting compact versions of numerous other formulae. 

Proving 0! is equal to 1!

According to the factorial notation formula;

n!=n×(n−1)! , where n=1

1! = 1× (1-1)!

1! = 1× 0!

1! = 0!

Or we can write as, 0! = 1!

Hence Proved!

Factorial of Negative Numbers

The factorials for all negative integers will have their imaginary part as zero; therefore, they are the real numbers. As already discussed, factorial in practical life gives the ways of permutation. However, it is impossible to permute any object or element less than 0, i.e. negative. Therefore, we can conclude that the factorial notation for negative numbers is undefined, whereas it is well-defined for positive numbers.

Application of Factorial Notation

The main application of Factorial notation is in permutation and combination.

• Permutation:  nPr=n!n-r!

Example of Permutation: Suppose you need to open a lock with the combination of any three numbers between 1 to 30. How many permutations will be possible without the repetition of numbers?

Solution: According to the question, there is a total of 30 digits with a combination of 3 digits as the code. Thus, the possibility will be 30P3 ways.

6P3 = 30! / (30-3)!

 = 30 × 29 × 28 × 27! / 27!

 = 30 × 29 × 28 [27! Cancels out]

 = 24360

Answer: Thus, there will be 44460 possible codes to open the lock.

• Combination:  :  nCr=n!r!n-r!

Example of Combination: There are five people in a group. In order to make two pairs for the dance combination, one person needs to be removed. The possible combinations shall be:

Solution: According to the formula:

6C4=5!2!5-2!

6C4=5!2!3!

=1202× 6

= 10

Sequences Comparable to Factorials

The comparable sequences with factorial in mathematics are as follows:

• Double-Factorials: These double factors have great importance in the trigonometry of mathematics.
• Multi-factorial: As the name suggests, these multi-factorials have multiple exclamation marks or points.
• Super-factorial: The super-factorials are the product of first n factorial terms. 
• Primorials: Primorials aims in getting the product of prime numbers (numbers that are not divisible by other numbers). These prime numbers are either less than or equal to n.
• Hyper-factorials: It is the result after multiplying consecutive values (from 1 to n numbers).

Miscellaneous Questions on Factorial Notation

Q 1- Find the value of 10!4!×5!

Solution:

10×9×8×7×6×5!4×3×2×1× 5!

=1260 

Q 2- Calculate the number of ways in which nine students can line up from left to right in a class photo.

Solution:

The number of ways in which the nine students shall line up will be equal to the factorial of the total number, i.e. 9.

= 9!

 = 9× 8 × 7 × 6 × 5 × 4 × 3 × 2!

=181440× 4

 =362880

Therefore, nine students can line up in 362880 ways.

Q 3- Determine the value of 10!2!×7!

Solution:

= 10×9×8×7!2×1× 7!

=360 

Q 4- What will be the value of 35!?

Solution:  The factorial of 35 will be as follows:

35× 34× 33× 32× 31× 30× 29× 28× 27× 26× 25× 24× 23× 22× 21× 20× 19× 18× 17× 16× 15× 14× 13× 12× 11× 10!

=10333147966386144929666651337523200000000

Conclusion 

According to the factorial notation definition, n! will be equal to the product of n and all its positive preceding numbers. For example, if we have 5! then this means it shall be written as 5×4×3×2×1. A French mathematician Christian Kramp in the early 1800, introduced this method as n!. Factorial notation is the basis for a variety of mathematical operations. For example, algebra, geometry, statistics, discrete mathematics, probability, permutation and combination. The factorial notation formula shall be n!= n×(n−1)!. The factorial for 0 is always equal to 1, as proved above. The factorial notation for negative numbers is undefined, whereas it is well-defined for positive numbers.