Permutations and Combinations Explained Simply
Permutations and combinations are the foundation of counting problems in maths. They tell you how many ways things can be arranged or selected, and they appear in Class 11 boards, JEE, and almost every aptitude exam. The key idea is simple: permutations care about order, combinations do not.
Factorial: the building block
The factorial of a whole number n, written n!, is the product of all whole numbers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition 0! = 1, which is a rule worth memorising. If remembering such facts is hard, try our tips on how to memorise maths formulas.
What is a permutation?
A permutation is an arrangement where order matters. The number of ways to arrange r items chosen from n distinct items is:
- nPr = n! / (n − r)!
Worked example: In how many ways can 3 books be arranged on a shelf from 5 different books? Here n = 5 and r = 3, so 5P3 = 5! / 2! = 120 / 2 = 60. Because the order on the shelf matters, "Book A then B" is different from "Book B then A", giving 60 arrangements.
What is a combination?
A combination is a selection where order does not matter. The number of ways to choose r items from n distinct items is:
- nCr = n! / [r! (n − r)!]
Worked example: In how many ways can you choose 3 books from 5 to take on holiday? Now order is irrelevant, so 5C3 = 5! / (3! × 2!) = 120 / (6 × 2) = 10. There are only 10 selections, far fewer than the 60 arrangements above.
The link between them
Permutations and combinations are related by a neat formula: nPr = nCr × r!. Each combination of r items can itself be arranged in r! ways, which turns selections into arrangements. Check it: 5C3 × 3! = 10 × 6 = 60 = 5P3.
When to use which
- Use permutations when position, rank, or sequence matters: passwords, seating in a row, prizes for first and second place.
- Use combinations when you only pick a group: forming a committee, choosing a team, selecting questions to answer.
Committee example: From 4 people, choosing any 2 for a committee is a combination: 4C2 = 6. But choosing a president and a vice-president is a permutation because the roles differ: 4P2 = 12. Ask yourself, "Does swapping two choices create a genuinely new outcome?" If yes, use permutations.
Common mistakes to avoid
- Reading the question wrong: words like "arrange", "order", or "rank" signal permutations; "select", "choose", or "group" signal combinations.
- Forgetting 0! = 1: this keeps formulas consistent for the boundary cases.
- Repetition: the formulas above assume no item is repeated; different rules apply when repetition is allowed.
These counting tools reappear constantly in probability and binomial theorem, so a firm grip now pays off later. For deeper practice aimed at entrance tests, see our JEE Maths resources, and revisit the basics anytime in Class 10 Maths.
Frequently asked questions
What is the main difference between permutation and combination?
Permutations count arrangements where order matters, while combinations count selections where order does not matter. That is why nPr is always greater than or equal to nCr.
Why is 0! equal to 1?
Defining 0! as 1 keeps formulas like nCr = n! / [r!(n − r)!] valid when r equals n. There is exactly one way to arrange or choose nothing, so the value 1 makes sense.
How do I remember the nPr and nCr formulas?
Remember that combinations divide by an extra r! to remove duplicate orderings. So nCr = nPr / r!. Learning one formula lets you quickly derive the other.