Permutation and Combination Calculator
Counts permutations nPr and combinations nCr for n and r up to 1000, with or without repetition. Every digit is exact and the formula is worked through. How it works
Use the Permutation and Combination Calculator
About the Permutation and Combination Calculator
P(10, 3) = 720 and C(10, 3) = 120: there are 720 ways to arrange 3 of 10 items in order, and 120 ways to choose 3 when order does not matter. This permutation and combination calculator gives both counts for any n and r up to 1000, with or without repetition, and shows the formula with your numbers in it. For n up to 26 it also lists the selections themselves, up to 300 of each: P(3, 2) is AB, AC, BA, BC, CA and CB.
nPr and nCr for n = 5 to 12
| n | P(n, 2) | C(n, 2) | P(n, 3) | C(n, 3) | P(n, 4) | C(n, 4) |
|---|---|---|---|---|---|---|
| 5 | 20 | 10 | 60 | 10 | 120 | 5 |
| 6 | 30 | 15 | 120 | 20 | 360 | 15 |
| 7 | 42 | 21 | 210 | 35 | 840 | 35 |
| 8 | 56 | 28 | 336 | 56 | 1,680 | 70 |
| 9 | 72 | 36 | 504 | 84 | 3,024 | 126 |
| 10 | 90 | 45 | 720 | 120 | 5,040 | 210 |
| 11 | 110 | 55 | 990 | 165 | 7,920 | 330 |
| 12 | 132 | 66 | 1,320 | 220 | 11,880 | 495 |
P counts ordered arrangements and C counts unordered groups, so P(n, r) is always C(n, r) × r!.
Counts people look up
| What is counted | Formula | Count |
|---|---|---|
| 5 cards from a 52-card deck | C(52, 5) | 2,598,960 |
| 13 cards from a 52-card deck | C(52, 13) | 635,013,559,600 |
| 6 numbers from 49 | C(49, 6) | 13,983,816 |
| 5 numbers from 69 | C(69, 5) | 11,238,513 |
| 3 people from a group of 10 | C(10, 3) | 120 |
| First, second and third among 10 | P(10, 3) | 720 |
| Orders of 5 books on a shelf | P(5, 5) | 120 |
| 4-digit PIN, digits may repeat | 10^4 | 10,000 |
| 3-letter code from 26 letters, letters may repeat | 26^3 | 17,576 |
Choosing 3 from 10 under each counting rule
| Order matters | Repetition allowed | Formula | Count |
|---|---|---|---|
| Yes | No | P(10, 3) | 720 |
| Yes | Yes | 10^3 | 1,000 |
| No | No | C(10, 3) | 120 |
| No | Yes | C(12, 3) | 220 |
With repetition the combination count is C(n + r − 1, r), which is why the last row reads C(12, 3).
How to use this tool
- Type n, the number of items to choose from, and r, the number you pick. Both are whole numbers from 0 to 1000.
- Leave Without repetition selected if each item can be picked only once, or choose With repetition if it can be picked again.
- Read the permutations (order matters) and the combinations (order does not matter). Each card shows its working, and a result longer than 15 digits can be copied in full.
- For small counts, read C(n, r) straight off the Pascal’s triangle table under the results, which covers n from 0 to 10, or press List the arrangements to see the selections written out with the items named A, B, C and so on.
Frequently asked questions
- What is the difference between a permutation and a combination?
- A permutation counts ordered arrangements and a combination counts unordered groups. Picking 2 letters from A, B and C gives 6 permutations (AB, BA, AC, CA, BC, CB) and 3 combinations (AB, AC, BC), which List the arrangements writes out for n = 3 and r = 2. Each group of r items can be put in r! orders, so P(n, r) = C(n, r) × r!.
- What is 5 choose 2?
- 10. C(5, 2) = 5! ÷ (2! × 3!) = 20 ÷ 2 = 10. The same inputs give P(5, 2) = 5 × 4 = 20 ordered pairs.
- How many 5-card poker hands are there?
- 2,598,960. That is 52 choose 5: the order of the cards does not matter and no card repeats. Counting the order in which the cards are dealt gives P(52, 5) = 311,875,200.
- How many combinations are there in a lottery that draws 6 numbers from 49?
- 13,983,816. That is C(49, 6), so a single ticket matches all six numbers with a chance of 1 in 13,983,816.
- How do I count when items can repeat?
- Choose With repetition. Ordered picks become n^r: a 3-letter code from 26 letters has 26^3 = 17,576 possibilities. Unordered picks become C(n + r − 1, r): 3 scoops from 10 ice cream flavors, with repeats allowed, is C(12, 3) = 220.
- What are C(n, 0) and 0!?
- 0! is 1, so C(n, 0) = 1 and P(n, 0) = 1: there is exactly one way to choose nothing. The tool also gives C(0, 0) = 1.
Related Tools
View all toolsnCr and nPr Calculator
Enter n and r up to 170 and choose combinations (nCr) or permutations (nPr) to get the exact count with its factorial working: C(52, 5) = 2,598,960.
Linear Regression Calculator
Least-squares line y = mx + b for your (x, y) data, with Pearson r, R², the number of points used and predictions of y from x or x from y.
Standard Deviation
Standard deviation calculator for a list of numbers: population σ and sample s side by side, both variances, plus mean, median, minimum, maximum, range and sum.
Betting Odds Calculator
Convert betting odds between American, decimal and fractional formats, see the implied probability, and get the profit and total return on a stake.
Average Calculator
Calculate mean, median, mode, range and sum for a list of numbers separated by commas, spaces or new lines, up to 10,000 values.
P-Value Calculator
P-value from a z, t, chi-square or F statistic and its degrees of freedom, left-, right- or two-tailed, with an optional comparison against alpha.