FACTORIAL

The factorial of a number can be defined as the function that multiplies the number by every natural number below it. Symbolically, factorials can be represented as “!”. So, n factorial is the product of the first n natural number and is represented as n!.

So n! Or “n factorial” means: n! = 1.2.3…n 

 product of first n positive integers = n(n – 1)(n – 2)… … … …3.2.1.

FACTORIAL FORMULA:

The factorial of a number is 

n! = n × (n-1) × (n-2) × (n-3) × …. × 3 × 2 × 1

For an integer n ≥ 1, the factorial representation in terms of ∏ product notation is:

n! =∏i=1ni

From the above formulas, the recurrence relation for the factorial of a number is the product of the factorial number and factorial of that number minus 1. It is denoted by:

n! = n. (n-1)!

 SUB FACTORIAL OF A NUMBER

A mathematical term “sub-factorial”, defined by the term “!n”, represents the number of arrangements of n given objects. It means the number of permutations of n objects so that no object stands in its original position.

FACTORIAL OF ZERO:

The factorial of 0 is 1, symbolically represented by  0! = 1.

• • when n = 0, then n!  is a product that involves the product of no numbers at all. An example of the, a product of no factors, is equal to the multiplicative identity.
• There is exactly one permutation of 0 objects.

FACTORIAL OF 5:

Finding the factorial of 5 is simple. This can be found using formulas and the expansion of numbers. It has been discussed step-by-step below-

We already know that 

n! = 1 × 2 × 3 …… × n

Factorial of 5 can be calculated as

5! = (1 × 2 × 3 × 4 × 5) = 120

Hence, 5!   is 120.

FACTORIAL OF 10:

For factorial of 10 is written as 10! = 10.9!

10! = 10(9 × 8 ×7 × 6× 5× 4× 3 × 2 × 1)

10! = 10(362880)= 3628800

Hence, 10!  is 3628800.

Factorial of numbers 1 to 10 table:

The list of factorial values from 1 to 10 is:

FACTORIAL OF NEGATIVE NUMBERS:

Can we have factorials of numbers like -1,-2 etc? Lets start with 3! = 3*2*1 = 6

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

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

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

(-1)! = 0! /0 = 1/0 =  undefined

And from here on down the entire integer factorials are undefined. So, the negative integer factorial is undefined.

SOME APPLICATIONS OF FACTORIAL VALUE:

Some applications in mathematics are as follows:

1. Recursion: 

A number can be represented in an expression by-

p! = p * (p-1) * (p – 2) * (p – 3)……… (p -(p-2)) * (p-(p-1)).

1. Permutations:

Arrangement of given r objects out of total n objects where the order is strictly important.

1. Combination:

Arrangement of given r objects out of total n objects when order is not important.

• Probability Distributions:

There are various probability distributions which include binomial distribution. The probability of an event can be calculated with the concept of permutations and combinations.

• Number Theory:

Factorial values are used extensively in number theory and also for approximation.

Factorials also occur in algebra through the binomial theorem and in calculus. Factorial is found in the theories of probability and numbers and even can be used to enable the manipulation of expressions.

PROPERTIES OF FACTORIAL:

The different properties of factorial are as follows:

• Growth and Approximation
• Divisibility 
• Continuous interpolation and non-integer generalization
• Computation.

Other Sequences similar to the Factorials:

In Mathematics, many sequences are comparable to the factorial. They include:

• Double Factorial: Which are used to simplify trigonometric integrals
• Multi- Factorials: This can be denoted with multiple exclamation points.
• Promorials:  The product of the prime numbers, which are less than or equal to n.
• Super – Factorials: which are defined as the product of first n factorials
• Hyper- Factorials: which are the results of multiplying the number of consecutive values ranging from 1 to n.

CONCLUSION:

The factorial of a number has many and intensive uses in permutations, combinations, and the computation of probability. We represent it by an exclamation mark (!). Factorials are also used in number theory, approximations, and statistics. In this topic, we discussed the Factorial Formula. We shall also learn the various applications of factorial formulas such as permutations, combinations, probability distribution.