AdvancedIn-depth4 min readUpdated 2 Oct 2026

What is the difference between permutations and combinations, and how do you know which to use?

SHORT ANSWER

Use permutations when order matters, like choosing a captain and vice-captain, and combinations when order does not matter, like choosing a committee. nPr = n! / (n - r)! and nCr = n! / (r! x (n - r)!). Each combination of r items can be arranged in r! ways, so nPr = nCr x r!.

Explain it simply

Ask one question: does swapping two chosen items give a different result?

Picking a captain and vice-captain from 5 players: Rohit as captain and Virat as vice-captain is different from the other way round. Order matters, so it is a permutation.
Picking 2 players to attend a press conference: Rohit and Virat is the same pair as Virat and Rohit. Order does not matter, so it is a combination.

A quick memory trick: Permutation goes with Position (arrangement), Combination goes with Choosing (selection).

Example

From 5 students, how many ways can you:

Choose a captain and a vice-captain? 5 choices for captain, then 4 for vice-captain: 5P2 = 5 x 4 = 20.
Choose a team of 2? Each pair was counted twice above, so 5C2 = 20 / 2 = 10.

Question: A committee of 3 men and 2 women is to be formed from 6 men and 5 women. In how many ways?

Men: 6C3 = (6 x 5 x 4) / (3 x 2 x 1) = 20.
Women: 5C2 = (5 x 4) / (2 x 1) = 10.
Both choices happen together, so multiply: 20 x 10 = 200.

Question: In how many ways can a team of 11 be picked from 15 players?

15C11 = 15C4 = (15 x 14 x 13 x 12) / (4 x 3 x 2 x 1) = 32,760 / 24 = 1,365.

const fact = (n) => (n <= 1 ? 1 : n * fact(n - 1));
const nPr = (n, r) => fact(n) / fact(n - r);
const nCr = (n, r) => fact(n) / (fact(r) * fact(n - r));
console.log([nPr(5, 2), nCr(5, 2), nCr(15, 11)]); // Output: [20,10,1365]

How it works

Fundamental counting principle: if one task can be done in m ways and the next in n ways, both can be done in m x n ways. "And" means multiply, "or" means add.
nPr = n! / (n - r)! counts ordered arrangements of r items out of n.
nCr = n! / (r! (n - r)!) counts unordered selections. Shortcut: nCr = nC(n - r), so 15C11 = 15C4.
Repeated letters: arrangements of a word = n! / (p! q! ...) where p, q are the counts of repeated letters. "APPLE" has 5! / 2! = 60 arrangements. "BANANA" has 6! / (3! x 2!) = 720 / 12 = 60.
Circular arrangements: n people around a round table can sit in (n - 1)! ways, because rotating everyone gives the same arrangement. 6 people: 5! = 120.
Items that must be together: tie them into one block, arrange the blocks, then arrange inside the block.

Real-world use

Permutations count things like possible 4-digit UPI PINs (10^4 = 10,000 with repetition allowed) or seating plans. Combinations count lottery tickets, possible fantasy cricket teams on Dream11, or ways to choose subjects for electives. This topic appears in most placement aptitude rounds and in CAT.

Common mistakes

Using permutation when the question only asks to "select" or "choose".
Forgetting to divide by the factorials of repeated letters.
Using n! instead of (n - 1)! for a round table.
Adding when you should multiply. Choosing men and women means multiply.
Ignoring whether repetition is allowed. A PIN with repetition gives 10^4 options, not 10P4.

What to say in the interview

"If swapping two chosen items gives a different outcome, order matters and I use permutations, nPr = n! / (n - r)!. If it gives the same outcome, like picking a committee, I use combinations, nCr = n! / (r! (n - r)!). They are linked by nPr = nCr x r!, because each group of r can be arranged in r! orders. For independent steps I multiply, for separate cases I add, and I watch for repeated letters, round tables and whether repetition is allowed."

Follow-up questions

How many ways can the letters of "MISSISSIPPI" be arranged?
10 people at a party each shake hands with every other person once. How many handshakes?
How many 4-digit numbers can be formed from the digits 1 to 9 with no digit repeated?
In how many ways can the letters of "LEADING" be arranged so that the vowels are always together?

Go deeper

Read these once the basics above feel easy.

Answers to the follow-ups:

11 letters with M = 1, I = 4, S = 4, P = 2: 11! / (4! x 4! x 2!) = 39,916,800 / 1,152 = 34,650.
Each handshake is an unordered pair: 10C2 = 45.
Order matters and no repetition: 9 x 8 x 7 x 6 = 3,024.
"LEADING" has 7 different letters, with vowels E, A, I. Treat the vowels as one block: 4 consonants + 1 block = 5 units, arranged in 5! = 120 ways. The vowels inside the block can be arranged in 3! = 6 ways. Total = 120 x 6 = 720.

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