Factorial Formula

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 will discuss the Factorial Formula with examples. We shall also learn the various applications of factorial formula such as permutations, combinations, probability distribution, etc. Let us start!

Definition of Factorial

The factorial formula is used to find the factorial of any number. It is defined as the product of the number with all its successive lowest value numbers till 1. Thus it is the result of multiplying the descending series of numbers. It must be remembered that the factorial of 0 is 1. Factorial Formula has many direct and indirect applications in permutations and combinations for probability calculation.

There are various functions based on factorials like double factorial, multifactorial, etc. Also, the Gamma function is an important concept based on factorial.

Formula for the Factorial

To get the factorial of a given number n the following given formula can be used,

\(p! = p \times (p-1) \times (p-2)… \times 2 \times 1\)

Also, the above computation can be done as follows:

For a number n, the factorial of it can be written as,

\(p! = p \times (p-1)!\) Or \(p! = p \times (p-1) \times (p-2)!\)

This is possible due to the recursive nature of factorial computation.

Let us understand it with some examples.

\(\frac{45!}{43!}\)

\(= \frac{43! \times 44 \times 45}{43!} = 44 \times 45\)

= 1980

Some Applications of Factorial Value

Some applications of factorial in mathematics are as follows:

1)Recursion

In the recursive definition of a number, we may use factorial. A number can be expressed in an expression containing the number only.

\(p! = p \times (p – 1) \times (p – 2) \times (p – 3) .. (p-(p – 2)) \times (p – (p – 1))\)

2) Permutations

Arrangement of given r things out of total n things when order is strictly important.

\(P_{r}^{n}=\frac{n!}{(n-r)!}\)

3) Combinations

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

\(C_{r}^{n} = \frac{n!}{(n-r)!r!}\)

4) Probability Distributions

There are various probability distributions like binomial distribution which include the use of factorial. To find the probability of an event, the concept of permutations and combinations is used a lot.

5) Number Theory

Factorials value are used extensively in number theory and also for approximations.

Solved Examples for Factorial Formula

Q.1: What is the value of 8!?

Solution: The formula for factorial is,

\(p! = p \times (p-1) \times (p-2) \times (p-3)… \times 2 \times 1 \\\)

\(8! = 8 \times (8-1) \times (8-2)… \times 2 \times 1 \\\)

\(= 8 \times 7 \times 6 \times …. 3 \times 2 \times 1\)

= 40320

Q.2: What is \(\frac {9!} {5!}\)?

Solution: The formula for factorial is,

\(p! = p \times (p-1) \times (p-2)… \times 2 \times 1 \\\)

Thus putting p =9

\(9! = 9 \times (9-1) \times (9-2)… \times 2 \times 1 \\\)

Similarly, Putting p = 5

\(5! = 5 \times (5-1) \times (5-2)… \times 2 \times 1 \\\)

Thus, \(\frac{9!}{5!} = 6 \times 7\times 8 \times 9\)

\(\frac{ 9! }{ 5! } = 3024\)