Permutations and Combinations Table
A permutation counts the number of ways to arrange r items chosen from n, where order matters. A combination counts the same selection without regard to order. This page provides quick-lookup tables for both, plus a calculator for any n and r.
Permutations & Combinations Calculator
Enter n (total items) and r (items chosen) to find nPr and nCr.
P(10, 3)720
C(10, 3)120
Formulas
Both formulas build on the factorial:
- Permutations: P(n, r) = n! / (n − r)!
- Combinations: C(n, r) = n! / (r! (n − r)!) = P(n, r) / r!
Permutations Table — P(n, r)
Number of ordered arrangements of r items chosen from n. A dash (—) means r > n, which is not possible.
| n | r=0 | r=1 | r=2 | r=3 | r=4 | r=5 |
|---|---|---|---|---|---|---|
| 0 | 1 | — | — | — | — | — |
| 1 | 1 | 1 | — | — | — | — |
| 2 | 1 | 2 | 2 | — | — | — |
| 3 | 1 | 3 | 6 | 6 | — | — |
| 4 | 1 | 4 | 12 | 24 | 24 | — |
| 5 | 1 | 5 | 20 | 60 | 120 | 120 |
| 6 | 1 | 6 | 30 | 120 | 360 | 720 |
| 7 | 1 | 7 | 42 | 210 | 840 | 2,520 |
| 8 | 1 | 8 | 56 | 336 | 1,680 | 6,720 |
| 9 | 1 | 9 | 72 | 504 | 3,024 | 15,120 |
| 10 | 1 | 10 | 90 | 720 | 5,040 | 30,240 |
Combinations Table — C(n, r)
Number of unordered selections of r items chosen from n. These are the same values as the binomial coefficients in Pascal's Triangle, which has the full triangle for higher rows.
| n | r=0 | r=1 | r=2 | r=3 | r=4 | r=5 |
|---|---|---|---|---|---|---|
| 0 | 1 | — | — | — | — | — |
| 1 | 1 | 1 | — | — | — | — |
| 2 | 1 | 2 | 1 | — | — | — |
| 3 | 1 | 3 | 3 | 1 | — | — |
| 4 | 1 | 4 | 6 | 4 | 1 | — |
| 5 | 1 | 5 | 10 | 10 | 5 | 1 |
| 6 | 1 | 6 | 15 | 20 | 15 | 6 |
| 7 | 1 | 7 | 21 | 35 | 35 | 21 |
| 8 | 1 | 8 | 28 | 56 | 70 | 56 |
| 9 | 1 | 9 | 36 | 84 | 126 | 126 |
| 10 | 1 | 10 | 45 | 120 | 210 | 252 |
Permutations vs. Combinations
- Order matters → permutation: arranging 3 books on a shelf, assigning 1st/2nd/3rd place in a race, forming a PIN code.
- Order doesn't matter → combination: choosing 3 toppings for a pizza, picking a 5-person committee, dealing a poker hand.
- A so-called "combination lock" is actually a permutation, since the order of the numbers matters.