Calculate nCr and nPr exactly: combinations and permutations with or without repetition, for poker hands, lottery odds and PIN codes.
By AdminDrafted with AI writing toolsUpdated Checked examples: 9
There are 2,598,960 ways to choose 5 of 52 items when order doesn't matter. The exact count has 7 digits. If all outcomes are equally likely, the probability of any one is 3.848 × 10⁻⁷.
Computed in exact BigInt arithmetic.
Assumes every outcome is equally likely.
Combinations count the ways to pick r items from n when order doesn't matter: C(n, r) = n!/(r!(n − r)!). Permutations count each ordering separately, P(n, r) = n!/(n − r)!, so P(n, r) = C(n, r) × r!. When an item can be picked more than once, ordered picks number nʳ and unordered picks C(n + r − 1, r), the "stars and bars" count. Every answer is an exact whole number, up to 20,000 digits. In boundary cases N denotes the count: choosing zero items gives N = 1; an impossible selection gives N = 0.
The default, 5 cards from 52, gives 2,598,960 poker hands, so one particular hand has a 1 in 2,598,960 chance. The same arithmetic gives 13,983,816 tickets in a 6-of-49 lottery, 720 podium orders from 10 runners and 10,000 four-digit PINs.
The chance of one outcome assumes every outcome is equally likely. Answers of 22 digits or more are shown in scientific notation, with every digit listed up to 300 digits.
A combination counts groups where order doesn't matter: nCr = n! ÷ (r!(n − r)!). A permutation counts ordered arrangements: nPr = n! ÷ (n − r)!. From 10 people there are C(10, 3) = 120 committees of three but P(10, 3) = 720 ways to award gold, silver and bronze.
Both formulas assume n different items, each used at most once. When items can repeat, as digits in a PIN do, use the repetition formulas further down.
n! (n factorial) multiplies every whole number from n down to 1: 5! = 5 × 4 × 3 × 2 × 1 = 120, and 8! = 40,320. By definition 0! = 1, because there is exactly one way to arrange nothing.
A factorial is a permutation of all n items, so n! = P(n, n). Factorials grow fast: 143! has 248 digits, and 200! has 375 digits, about 7.89 × 10³⁷⁴.
That is far past the largest double-precision number, about 1.8 × 10³⁰⁸, so software that stores numbers as doubles overflows at 171!. Exact integer arithmetic avoids that limit.
Cancel before you multiply. In C(10, 3) = 10! ÷ (3! × 7!), the 7! cancels the tail of 10!, leaving (10 × 9 × 8) ÷ (3 × 2 × 1) = 720 ÷ 6 = 120.
Choosing r items is the same as choosing the n − r to leave out, so C(n, r) = C(n, n − r). C(52, 47) equals C(52, 5) = 2,598,960, and the smaller r is quicker to work.
| n | r | Combinations nCr | Permutations nPr |
|---|---|---|---|
| 5 | 2 | 10 | 20 |
| 10 | 3 | 120 | 720 |
| 52 | 5 | 2598960 | 311875200 |
| 49 | 6 | 13983816 | 10068347520 |
| 8 | 0 | 1 | 1 |
| 8 | 8 | 1 | 40320 |
The 52-and-5 row is the number of five-card poker hands, and 49 and 6 is a 6-from-49 lottery. The last two rows show 0! = 1 at work: there is one way to choose nothing and one way to choose everything.
Multiply r numbers, counting down from n. A podium from 10 runners has 10 choices for gold, 9 for silver and 8 for bronze: 10 × 9 × 8 = 720.
The same three medalists can stand on the podium in 3! = 6 orders, which is why 720 is 6 × 120. In general nPr = nCr × r!.
Ask whether swapping two of the chosen items gives a different outcome.
Roles turn a group into an arrangement. Three of ten people as president, secretary and treasurer is P(10, 3) = 720, not 120. A "combination lock" is really a permutation lock, since 1-2-3 and 3-2-1 are different codes.
When an item can be chosen again, ordered picks number nʳ: each of the r positions has n options. A four-digit PIN has 10⁴ = 10,000 codes, from 0000 to 9999.
Unordered picks with repetition number C(n + r − 1, r), the "stars and bars" count. Three scoops from five flavors, repeats allowed, give C(7, 3) = 35 cups; chocolate, chocolate, vanilla counts once however the scoops are stacked.
| n | r | Unordered with repetition | Ordered with repetition |
|---|---|---|---|
| 5 | 3 | 35 | 125 |
| 10 | 4 | 715 | 10000 |
| 3 | 2 | 6 | 9 |
| 1 | 5 | 1 | 1 |
With only one kind of item, every pick is the same, so 1 and 5 gives a single outcome in both columns.
If every outcome is equally likely, the chance of one particular outcome is 1 ÷ count. A single 6-from-49 lottery ticket wins with probability 1 in 13,983,816.
For an event made of several outcomes, divide the favorable count by the total. Of the 2,598,960 possible hands, 4 are royal flushes, so the chance of being dealt one is 1 in 649,740.
For unions, overlaps and conditional chances, use the probability calculator. For dice totals and card draws, use the dice, coin and card calculator.
OpenStax's counting principles chapter works through the multiplication principle, permutations and combinations with more examples.
Checked against: C(52,5) = 2,598,960
Checked against: 10 × 9 × 8 = 720
Checked against: 1/13983816 = 7.15112e-8
Checked against: Independent calculation
A permutation counts orderings; a combination counts only which items are chosen. From 10 runners there are P(10, 3) = 720 ways to fill gold, silver and bronze, but only C(10, 3) = 120 different groups of three, because each group can be ordered in 3! = 6 ways. Use permutations for rankings, seatings and passwords, and combinations for hands, committees and lottery tickets.
There are C(52, 5) = 2,598,960 five-card hands from a standard 52-card deck, counting each set of cards once regardless of the order dealt. Only 4 of them are royal flushes, so the chance of being dealt one is 4 in 2,598,960, or 1 in 649,740. Counting ordered deals instead gives P(52, 5) = 311,875,200.
Multiply the number of combinations for each drawn set. A 6-from-49 draw has C(49, 6) = 13,983,816 possible tickets. Powerball draws 5 of 69 white balls and 1 of 26 red balls, so the jackpot odds are C(69, 5) × 26 = 11,238,513 × 26, or 1 in 292,201,338. Order doesn't matter in these games, which is why combinations apply.
There are 10⁴ = 10,000 four-digit PINs, from 0000 to 9999, because each of the 4 positions can hold any of 10 digits. Strictly this is a permutation with repetition, since 1234 and 4321 are different codes; a "combination lock" is really a permutation lock. A 6-digit PIN has 10⁶ = 1,000,000 possibilities.
0! = 1 by definition, because there is exactly one way to arrange zero items: the empty arrangement. The convention keeps the formulas consistent, so C(n, 0) = n!/(0! × n!) = 1 and C(n, n) = 1. It also matches the gamma function, where 0! = Γ(1) = 1, and the recursion n! = n × (n − 1)! at n = 1.
This calculator includes 9 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.