Combination Permutation Calculator
Calculate combinations (nCr) and permutations (nPr) for any n and r with exact integer results.
Combinations nCr
120
10 choose 3
Formula
n! ÷ (r! × (n−r)!)
Factorial notation
Combinations ignore order (teams, lottery picks); permutations count order (passwords, podium finishes).
Last updated: February 2026
How this calculator is verified
Checked by True Calculator automated test suite on
- Formula verified against a published worked example in the automated test suite
- Edge cases (zero, negative, boundary and unit-mismatch inputs) covered by unit tests
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
Use this calculator when a problem is really a counting question: how many ways to choose, arrange or rank items. Students preparing for aptitude tests and competitive exams in India meet these problems constantly, and a wrong mental multiplication is easy — 52C5 = 2,598,960 is not something anyone should trust to mental arithmetic. Organisers count possible team selections, event planners count seating arrangements, and anyone designing PINs, passwords or codes counts permutations to judge security. The two modes side by side also teach the core idea: permutations are combinations multiplied by r!, because ordering multiplies every selection. Whether the question comes from a textbook, an exam paper or a real scheduling problem, the calculator gives the exact integer in one step.
How to Use This Calculator
- Step 1: Enter the total number of items (n).
- Step 2: Enter how many items you are selecting or arranging (r).
- Step 3: Choose combination if order does not matter, or permutation if it does.
- Step 4: Read the exact integer result, and compare the two modes to see the order factor.
Worked Example
From 10 candidates you need to pick 3 for a team. Enter n = 10, r = 3 and choose combination: the calculator returns 10C3 = 10! ÷ (3! × 7!) = 120 possible teams. If the same 3 people must be arranged in a first-second-third ranking instead, choose permutation and the result jumps to 10P3 = 10! ÷ 7! = 720 arrangements. For a 5-card poker hand from a 52-card deck, 52C5 returns 2,598,960 — the denominator that makes strong hands so rare.
Tips and Common Mistakes
- •Tip 1: Use permutation for anything with ranks or order — podiums, passwords, seating plans.
- •Tip 2: Use combination for selections where order is invisible — teams, committees, lottery picks.
- •Tip 3: For n = r, permutations equal n! while combinations equal 1 — a good sanity check.
- ✗Mistake 1: Avoid entering r larger than n; the calculator rejects it because you cannot pick more items than exist.
- ✗Mistake 2: Don't use combinations when order matters — that mistake undercounts by a factor of r!.
Frequently Asked Questions
What is the difference between a combination and a permutation?
A combination ignores order while a permutation counts it. Choosing 3 people from 10 for a team is 120 combinations, but arranging 3 of them in a line is 720 permutations.
How do you calculate 10P3 and 10C3?
nPr = n! ÷ (n−r)!, so 10P3 = 10 × 9 × 8 = 720. nCr = n! ÷ (r! × (n−r)!), so 10C3 = 720 ÷ 6 = 120. The calculator applies the same formulas for any n and r.
Why is 52C5 such a large number?
Choosing 5 cards from a standard deck of 52 ignores order, and there are 2,598,960 distinct five-card hands. That is why poker probabilities are so small — the denominator is nearly 2.6 million.
What happens if I enter r greater than n?
You cannot choose or arrange more items than exist, so the calculator shows a prompt rather than a number. Keep r less than or equal to n.
Where are combinations and permutations used in real life?
Lottery odds, tournament scheduling, password and PIN security counts, seating arrangements and committee selection all reduce to nCr or nPr problems.
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