Permutation Formula | Formula, Examples and Practice Problems (2025)

Permutation Formula: In mathematics, permutation relates to the method of organizing all the members of a group into some series or design. In further terms, if the group is already completed, then the redirecting of its components is called the method of permuting. Permutations take place, in better or slightly effective methods, in almost every district of mathematics. They usually occur when different directions on detailed restricted sites are monitored.

What is the Permutation Formula?

It is the separate arrangements of a supplied numeral of associates taken one by one, or some, or all at a time. For instance, if we have two elements A and B, there are two possible interpretations, AB and BA.

An integer of permutations when ‘r’ elements are arranged out of a total of ‘n’ elements is,

ⁿP_r = n! / (n – r)!.

For example,

Let n = 2 (A and B) and r = 1 (All permutations of size 1). The answer is 2!/(2 – 1)! = 2. The two permutations are AB, and BA.

Permutation Formula Explanation

A permutation is a kind of arrangement that shows how to permute. If there are three separate integers 1, 2, and 3, and if somebody is interested to permute the integers taking 2 at a point, it offers (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), and (3, 2). That is it can be performed in 6 ways.

Here, (1, 2) and (2, 1) are separate. Again, if these 3 integers shall be set enduring all at a time, then the arrangements will be (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2) and (3, 2, 1) i.e. in 6 ways.

In known, n separate items can be selected accepting r (r < n) at a time in n(n – 1)(n – 2) … (n – r + 1) ways. In particular, the first item can be any of the n items. Now, after selecting the first item, the second item will be any of the remaining n – 1 thing. Similarly, the third item can be any of the remaining n – 2 things. Alike, the r^th item can be any of the remaining n – (r – 1) things.

Therefore, the total numeral of permutations of n separate items taking r at a time is n(n – 1)(n – 2) … [n – (r – 1)] which is noted as ⁿP_r. Or, in other words,

ⁿP_r = n!/(n – r)!

Sample Problems on Permutation Formula

Question 1: What are the types of permutations?

Solution:

The permutation of a collection of things or components in order relies on three conditions:

1. When recurrence of essences is not allowed
2. When recurrence of essences is allowed
3. When the components of a group are not different

Question 2: Calculate the number of permutations of n = 5 and r = 2.

Solution:

Given,

n = 5

r = 2

Using the formula given above:

Permutation: ⁿP_r = (n!) / (n – r)!

= (5!) / (5 – 2)!

See Also

Combinations and PermutationsPermutation ( Definition, Formula, Types, and Examples)Permutation - Formula, Definition, Examples, Permutations and CombinationsPermutation and Combination - Definition, Formulas, Examples and FAQs

= 5! / 3! = (5 × 4 × 3! )/ 3!

= 20

Question 3: How many 3 letter phrases with or without purpose can be created out of the letters of the word POEM when repetition of letters is not permitted?

Solution:

Here n = 4, as the word POEM has 4 letters. Since we have to create 3 letter words with or without meaning and without repetition, therefore total permutations possible are:

⇒ P(n, r) = 4!/(4 − 3)!

= 4 × 3 × 2 × 1/1

= 24

Question 4: How many 4 letter phrases with or without purpose can be created out of the letters of the word KANHA when repetition of words is permitted?

Solution:

The number of letters, in this case, is 5, as the word KANHA has 5 alphabets.

And r = 4, as a 4-letter term has to be selected.

Thus, the permutation will be:

Permutation (when repetition is permitted) = 5⁴= 625

Question 5: It is required to place 4 men and 3 women in a row so that the women entertain the even positions. How many such configurations are feasible?

Solution:

We are given that there are 4 men and 3 women.

i.e. there are 7 positions.

The even positions are: 2nd, 4th, and the 6th places

These three places can be occupied by 3 women in P(3, 3) ways = 3!

= 3 × 2 × 1

= 6 ways

The remaining 4 positions can be occupied by 4 men in P(4, 4) = 4!

= 4 × 3 × 2 × 1

= 24 ways

Therefore, by the Fundamental Counting Principle,

Total number of ways of seating arrangements = 24 × 6

= 144

Question 6: Find the number of phrases, with or without meaning, that can be comprised with the letters of the word ‘TABLE’.

Solution:

‘TABLE’ contains 5 letters.

Thus, the numeral of phrases that can be formed with these 5 letters = 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 7: Find the number of permutations of the letters of the phrase subject such that the vowels consistently appear in odd positions.

Solution:

The word ‘SUBJECT’ has 7 letters.

There are 6 consonants and 1 vowels in it.

No. of ways 1 vowels can occur in 7 different places = ⁷P₁ = 7 ways.

After 1 vowels take 1 place, no. of ways 6 consonants can take 6 places = ⁶P₆ = 6! = 720 ways.

Therefore, total number of permutations possible = 720 × 720 = 518,400 ways.

Articles related to Permutation Formula:

• Permutation and Combination
• Permutation – Formula, Definition, Examples
• How to Use nPr Formula with Examples
• Definition, Formula, Permutations and Combinations

Summary – Permutation Formula

The permutation formula helps you calculate the number of unique arrangements (order matters) for a selection of items from a larger set. It is denoted by nPr, where n is the total number of items and r is the number of items you are choosing to arrange. The formula is nPr = n! / (n – r)!, where n! represents n factorial (n multiplied by all the positive integers less than n).

Also Check:

• Permutation and Combination Class 11 Notes
• Permutation and Combination Class 11 NCERT Solutions

Practice Problems on Permutation Formula

1. There are 4 different colors to paint a car. How many unique color combinations are there if order matters (e.g., red-blue-green is different from green-blue-red)?

2. A club has 10 members and needs to choose a president and vice president. How many ways can they do this if order matters (i.e., president then vice president is different from vice president then president)?

3. A bookshelf has space for 8 books. How many unique ways can you arrange 5 different books on the shelf?

4. A password needs to be 6 characters long, created from the letters A-Z (uppercase only) and the digits 0-9. How many unique passwords are possible if repetition is allowed (same character can be used multiple times)?

5. A team needs to choose a captain and vice captain from 7 players. How many ways can they do this if order doesn’t matter (i.e., captain then vice captain is the same as vice captain then captain)?

6. A student needs to arrange 3 books out of 7 distinct books on a shelf. How many different arrangements are possible?

7. From a group of 8 people, how many ways can a committee of 3 be formed where the order of selection matters?

8. You have 5 different types of chocolates and need to create a gift box with 2 chocolates. How many different combinations of chocolates can you make if order matters?

9. How many distinct 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 if repetition of digits is allowed?

10. In a lottery, there are 12 different numbers. How many ways can you choose 4 numbers in sequence if order matters?

FAQs on Permutation Formula

What is the difference between permutations and combinations?

Permutations are arrangements where the order of items matters, while combinations are selections where the order does not matter. For example, arranging books on a shelf is a permutation problem, but selecting books for a reading list without caring about the order is a combination problem.

Can permutations be calculated for negative values of r?

No, permutations are defined only for non-negative values of r. If r is negative, the calculation is not defined.

How do permutations apply to real-life scenarios?

Permutations are used in various real-life situations such as arranging books on a shelf, scheduling tasks, determining possible outcomes in games or lotteries, and creating unique passwords.

What is the factorial notation used in permutations?

Factorial notation (denoted as n!) represents the product of all positive integers up to n. For example 5! = 5 X 4 X 3 X 2 X 1 = 120 It is used in the permutation formula to count the number of possible arrangements.

How do I calculate permutations when some items are identical?

When some items are identical, use the formula for permutations of a multiset: n! / n₁! . n₂! …n_k! where n is the total number of items, and n1 ,n2 …. nk are the frequencies of the identical items.