Permutation & Combination Calculator
Enter n (total items) and r (selected items) to compute both nPr and nCr.
Permutation vs Combination
Permutation nPr — order matters
nPr = n! / (n − r)! — Example: arranging 3 of 5 books: 5P3 = 60
Combination nCr — order does NOT matter
nCr = n! / (r! × (n − r)!) — Example: choosing 3 of 5 books: 5C3 = 10
Relationship
nPr = nCr × r! — Permutations are always ≥ combinations.
Related Calculators
Permutations vs Combinations — Key Difference
The only difference between permutations and combinations is whether order matters. If you're choosing a 3-digit PIN from digits 1–9, order matters (123 ≠ 321) — use permutations: P(9,3) = 504. If you're choosing 3 toppings for a pizza from 9 options, order doesn't matter (pepperoni+mushroom+olive is the same whatever order you list them) — use combinations: C(9,3) = 84. The formula is C(n,r) = P(n,r) / r!. The factorial in the denominator divides out the redundant orderings.
Worked Examples
| Scenario | Type | Formula | Answer |
|---|---|---|---|
| How many ways to arrange 5 books? | Permutation | 5! | 120 |
| Choose 2 from 10 for a committee | Combination | C(10,2) | 45 |
| 4-digit code from 0–9, no repeats | Permutation | P(10,4) | 5,040 |
| Lottery: pick 6 from 49 | Combination | C(49,6) | 13,983,816 |
When to Use Permutations vs Combinations
Use permutations when ORDER matters. The number of ways to arrange r items from n distinct items is P(n,r) = n!/(n−r)!. Arranging 3 runners on a podium from 10: P(10,3) = 10×9×8 = 720 ordered arrangements. Use combinations when ORDER does NOT matter. Choosing 3 players from 10 for a team: C(10,3) = 10!/(3!×7!) = 120 groups. The key question: does swapping two selected items create a different outcome? If yes → permutation. If no → combination.
Permutations: ranking 5 books on a shelf, assigning roles in a play, scheduling ordered tasks, PIN code possibilities. Combinations: choosing committee members, selecting lottery numbers, picking a hand of cards, sampling without replacement. Note: C(n,r) = P(n,r)/r!, because combinations don't count the r! orderings of each selection. C(10,3) = 720/6 = 120.
P(n,r) vs C(n,r) Comparison
| n | r | P(n,r) | C(n,r) | Ratio (r!) |
|---|---|---|---|---|
| 5 | 2 | 20 | 10 | 2 |
| 6 | 3 | 120 | 20 | 6 |
| 10 | 3 | 720 | 120 | 6 |
| 52 | 5 | 311,875,200 | 2,598,960 | 120 (cards) |
Permutations and Combinations in Probability
Permutations and combinations both count ways to select items from a set, but they differ in one critical respect: permutations account for order, while combinations do not. If you are selecting 3 runners for first, second, and third place from a group of 10, the order matters and the count is a permutation. If you are selecting 3 committee members from a group of 10 where all three roles are equal, order does not matter and the count is a combination. The permutation formula is n factorial divided by (n minus r) factorial, where n is the total number of items and r is the number selected. The combination formula is n factorial divided by the product of r factorial and (n minus r) factorial. The combination count is always less than or equal to the permutation count for the same n and r because each combination corresponds to r factorial different permutations. These formulas underpin probability theory, statistical sampling, lottery odds, experimental design, and combinatorial optimization in computer science. The notation C(n, r) or the binomial coefficient appears throughout the binomial theorem and Pascal's triangle.
Permutation and Combination Reference Table
| n | r | Permutations P(n,r) | Combinations C(n,r) |
|---|---|---|---|
| 5 | 2 | 20 | 10 |
| 6 | 3 | 120 | 20 |
| 8 | 2 | 56 | 28 |
| 10 | 3 | 720 | 120 |
| 10 | 5 | 30240 | 252 |
