The product of a given whole number ‘n’ with every whole number until 1 is called the factorial. The function factorial can be defined as the function that multiplies itself by each natural number below it. It is essentially denoted by an exclamation mark ‘!’. The factorial notation is given as n! (n factorial). The factorial function is used to calculate the number of ways in which “n” objects can be organised. In a mathematical sense, it can be represented as

n!=n .(n-1) . (n-2) ….2 . 1

n!=n . (n-1)!

The older notation of factorial was represented as Π(n). 

Zero Factorial (0!)

The factorial of 0 is 1. In the mathematical form, it is 0!=1. The various explanations for this result are:

• Zero objects will be one permutation since there is nothing to rearrange. Hence the sole option is to do nothing. 
• The definition of n! as a product for n=0 entails the product of no integers at all. So is an example of the broader tradition that the empty product, or a product with no factors, is equivalent to the multiplicative identity.
• In combinatorics, the field of mathematics based on counting, the definition also validates many identities, which explains the above result.

Factorial of Negative Numbers

Factorial of negative integers is always undefined. To explain this lets take 3! = 3 x 2 x 1 = 6 and proceed further. 

2! = 3!/3 = 6/3 = 2

1! = 2!/2 = 2/2 = 1

0! = 1!/1 = 1/1 = 1

(-1)! = 0!/0 = 1/0

(any number divided by 0 is undefined)

All integer factorials are undefined from here on out. Negative integer factorials, then, are undefinable.

Use of Factorial

Factorial is mainly used in Permutations and Combination. A permutation can be expressed as an arrangement of things in which the order matters. In other words, the number of ways of selection and arrangements of items in which order matters. It can be calculated using the formula 

n Pr= n!(n-r)!

Here, n is represented as the things, and r represents the things taken at a time.

The combination is the process of grouping outcomes in which the order does not matter. It can also be defined as the number of ways of selecting items in which order of grouping is not important. It is formulated as 

n Cr= n!r!(n-r)!

Here, n is represented as the things, and r represents the things chosen at a time.

Factorial is also used in the coefficients of terms of the binomial expansion.

History of Factorial

One of the first known accounts of factorials in Indian mathematics comes from the Anuyogadvara-sutra, one of the canonical works of Jain literature, with dates ranging from 300 BCE to 400 CE. It distinguishes the sorted and reversed order of a group of items from the other (“mixed”) orders, calculating the number of mixed orders by subtracting two from the factorial’s typical product calculation. In 1808, the French mathematician Christian Kramp invented the notation n! for factorials. There have also been a variety of additional notations used. Another subsequent notation, in which the factorial argument is half-enclosed by the left and bottom sides of a box, was popular in Britain and America but has since fallen out of favour, possibly due to its difficulty typeset. The term “factorial” (originally French: factorielle) was used by Louis François Antoine Arbogast in his first study on Faà di Bruno’s formula in 1800. Still, it referred to a broader concept of arithmetic progression products. The “factors” in this name allude to the factorial’s product formula terms.

Applications of Factorial

Applications of factorial are as follows:

1. Probability Distributions: Other probability distributions, such as the binomial distribution, use factorial. Permutations and combinations are frequently used to calculate the probability of an event.
2. Recursion: In the recursive definition of a number, we may use factorial. A number in an expression can be expressed containing the number only.
3. Permutations: Arrangement of given r things out of total n objects given that the order is strictly important.
4. Combinations: Arrangement of given r things out of total n things when order is unimportant.
5. Number Theory: Factorial values are used extensively in approximations and number theory.

Factorial Table

The factorial table below shows us the factorial n values for the first 10 integers.

Questions based on Factorial n

1. In how many distinct ways can 7 people come 1st, 2nd and 3rd?

Solution:

Let the 7 people be called a, b, c, d, e, f and g, then the includes: abc, abd, abe, abf, abg, acb, ace, acf… etc. The formula is 7!(7-3)!= 7!4! = 7x6x5x4x3x2x14x3x2x1= 7x6x5 = 210

Therefore, there are 210 discrete ways that 7 people could come 1st, 2nd and 3rd.

  2. How many 5-digit numbers can be made using the digits 1, 2, 5, 7, and 8 without any digit being repeated?

Solution:

The given digits (1, 2, 5, 7 and 8) should be arranged among themselves in order to get 5 digit numbers. The enumerate ways in which this can be done using the factorial.

5!= 5 x 4 x 3 x 2 x 1=120

Therefore, the required number of 5- digit numbers is 120.

3. How many distinct ways can the letter in the word “document” be arranged?

Solution:

To solve this problem, we simply multiply the number of letters in the word by the factorial. This works since each letter in the word is distinct, and we’re merely calculating the maximum number of possible combinations for ordering eight goods.

8!=8x7x6x5x4x3x2x1= 40,320

Conclusion

The factorial is represented as n!. It is stated for a positive integer as the product of all the positive integers preceding to n. The factorial function has its use in various aspects in the field of mathematics like number theory, probability and statistics, permutations and combinations etc. We also learned about the special case of zero factorial, which always equals 1. The basic definition of the factorial in a mathematical sense is that there are n! number of ways in which n distinct objects can be arranged into an ordered sequence.