Permutation Formula
The permutation formula is used to find the different number of arrangements that can be formed by taking r things from the n available things. Permutations are useful to form different words, number arrangements, seating arrangements, and for all the situations involving different arrangements. As per the permutation formula, the permutation of 'r' objects taken from 'n' objects is equal to the factorial of n divided by the factorial of difference of n and r.
What is Permutation Formula?
In the below formula of permutations, the total number of things are 'n' and 'r' number of things can be selected. Before applying the permutation formula, we need to clearly know the concept of factorial. The factorial of any number is the product of the consecutive numbers starting from 1, and till that number. n! = 1 × 2 × 3 × 4 × .......× n. Let us now check the below permutation formula.
\[^nP_r = \dfrac{n!}{(n  r)!}\]
Let us see the application of the permutation formula in the following solved examples.
Solved Examples on Permutation Formula

Example 1: Find the number of ways in which a threedigit code can be formed, using the numbers from the digits 4, 5, 7, 8, 9.
Solution:
The given digits are 4, 5, 7, 8, 9.
Total number of digits = n = 5
The number of digits in the code = r = 3
Applying the permutation formula we have:
The number of ways of forming a 3 digit code =\(^5P_3 \)
\[\begin{align} ^5P_3 &= \frac{5!}{(5  3)!} \\&= \frac{5!}{2!} \\&= \frac{5 \times 4 \times 3 \times 2 \times 1 }{2 \times 1} \\&=5 \times 4 \times 3 \\&= 60 \end{align}\]
Answer: Hence, there are 60 ways of forming a threedigit code. 
Example 2: How many different words with or without meaning, can be formed using any 4 letters from the word 'Dictionary'?
Solution:
The given word is 'Dictionary'.
Total number of digits = n = 10
Number of alphabets to form the new word = r = 4
Using the permutations formula we have:
Total number of 4 letter words which can be formed = \(^{10}P_4\)
\[\begin{align} ^{10}P_4 &= \frac{10!}{(10  4)!} \\&= \frac{10!}{6!} \\&= \frac{10 \times 9 \times 8 \times 7 \times 6! }{6!} \\&=10 \times 9 \times 8 \times 7 \\&= 5040 \end{align}\]
Answer: Hence a total of 5040 fourletter words can be formed.