Permutation Formula

The permutation is the arrangement of objects in some definite order. Do you wonder how the schedules of trains and busses are made to suit our convenience? The concept of permutation and the permutation formula comes to the rescue. Also when we see the number plates of vehicles which consist of few alphabets and digits. We can easily prepare these codes using permutations. In this article, we will see the definition of permutation and permutation formula. Let us see the concept.

Permutation Formula

What is Permutation?

A permutation is a very important computation in mathematics. It is an arrangement of all or part of a set of objects, with regard to their order of the arrangement. Actually, very simply put, a permutation is an arrangement of objects in a particular way.

While dealing with permutation we should concern ourselves with the selection as well as the arrangement of the objects. Thus, ordering is very much essential in permutations.

For example, suppose we have a set of three letters: P, Q, and R. We have to find the number of ways we can arrange two letters from that set. Each possible arrangement will be one example of permutation. The complete list of possible permutations is PQ, PR, RP, QR, RP, and RQ.

When they refer to permutations, mathematicians use specific terminology. They describe permutations as an event when n distinct objects taken r at a time. Here, translation n refers to the number of objects from which the permutation is formed. Also, r refers to the number of objects used to form the permutation.

Consider the example given above. The permutation was formed from 3 alphabets (P, Q, and R),

So, n = 3;

Permutation consisted of 2 letters,

so r = 2.

Permutation Formula

The number of permutations of n objects, when r objects will be taken at a time.

nPr=(n) × (n-1) × (n-2) × …..(n-r+1)

i.e. nPr =\(\frac{n!}{(n-r)!}\)

Here n! is the Factorial of n. It is defined as:

n!= (n) × (n-1) × (n-2) ×…..3 × 2 × 1

Other notation used for permutation: P(n,r)

In permutation, we have two main types as one in which repetition is allowed and the other one without any repetition. And for non-repeating permutations, we can use the above-mentioned formula.

For the repeating case, we simply multiply n with itself the number of times it is repeating. It means that \(n^r\), where n is the number of things to be chosen from and r, is the number of items being chosen.

Solved Examples

Q. How many 3 letter words with or without meaning can be framed out of the letters of the word SWING? Repetition of letters is not allowed?

Solution: Here n = 5, because the number of letters is 5 in word SWING.

Since we have to frame words of 3 letters without repetition.

Therefore permutations possible are:

P(n,r)

= \(\frac{5!}{(5-3)!}\)

= 60

Q. How many 3 letter words with or without meaning can be created out of the letters of the word SMOKE. Note that the repetition of letters is allowed?

Solution:

The number of objects, here is 5, because the word SMOKE has 5 alphabets.

Also, r = 3, as 3 letter-word has to be chosen.

Thus the permutation will be

= \(n^r\) as repetition is allowed.

= \(5^3\)

= 125