Permutation & Combination Calculator
Calculate nPr (permutations) and nCr (combinations) for any n and r.
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Enter n (the total number of items) and r (how many you pick) and this calculator returns the number of combinations (nCr), the number of permutations (nPr) and n factorial. It covers the counting questions in Class 11 maths, probability homework and the aptitude sections of exams like CAT and bank PO tests.
The two formulas
nPr = n! ÷ (n − r)! nCr = n! ÷ (r! × (n − r)!) = nPr ÷ r!
For n = 10 and r = 3: 10P3 = 10 × 9 × 8 = 720, and 10C3 = 720 ÷ 3! = 720 ÷ 6 = 120. The third line shows 10! = 3,628,800. Combinations are always smaller than or equal to permutations, because each group of r items can be arranged in r! different orders.
Does order matter? Choosing the right one
| Question | Use | Answer |
|---|---|---|
| Pick a president, vice-president and secretary from 10 members | 10P3 | 720 |
| Pick a 3-person committee from 10 members | 10C3 | 120 |
| Choose a playing XI from a 15-player squad | 15C11 | 1,365 |
| Arrange the XI into a batting order | 11P11 | 39,916,800 |
| Pick 6 numbers from 49 in a lottery | 49C6 | 13,983,816 |
| Deal a 5-card hand from 52 cards | 52C5 | 2,598,960 |
Ask yourself whether swapping two chosen items creates a different outcome. If a committee of Asha, Ben and Chitra is the same as Chitra, Ben and Asha, order does not matter and you need nCr. If Asha as president differs from Asha as secretary, you need nPr.
Limits of the calculator
- n can be at most 170, because 171! is larger than the biggest number standard floating-point arithmetic can hold (about 1.8 × 10³⁰⁸).
- Very large answers are approximate. Results are exact up to roughly 15 or 16 significant digits. 100C50, for example, shows the correct leading digits followed by zeros, while the exact value ends in …497,256.
- r must be between 0 and n, and decimals are rounded down. Since 0! = 1, choosing 0 items gives exactly one combination.
Counting problems this does not cover directly
- Repetition allowed: a 4-digit PIN where digits can repeat has 10⁴ = 10,000 possibilities, calculated as nʳ rather than nPr. The PIN generator creates such codes.
- Combinations with repetition: picking 3 scoops from 5 ice-cream flavours, repeats allowed, is C(n + r − 1, r). Enter n = 7 and r = 3 to get 35.
- Identical items: the letters of BANANA can be arranged in 6! ÷ (3! × 2!) = 60 distinct ways, dividing out the repeated A's and N's.
- Circular seating: n people around a round table can sit in (n − 1)! ways, since rotations count as the same arrangement.
If you need to actually split people into random groups rather than count the possibilities, the team generator does that.