Permutation & Combination Calculator

Enter n and r — get nPr and nCr instantly, with and without repetition, the exact formula used, and a plain-English reading of what each number counts.

Values

Combinations

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nCr (no repetition)—
nPr (no repetition)—
Combinations with repetition—
Permutations with repetition (nʳ)—
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Related calculators

Percentage Fraction Average / mean Ratio Standard deviation Exponent

Permutations vs combinations: does order matter?

Both count the ways to pick r things from n. The difference is whether rearranging the picks counts as new. Choosing gold, silver and bronze from 8 runners is a permutation — Ana-Ben-Cy is a different podium from Ben-Ana-Cy — so there are 8P3 = 8 × 7 × 6 = 336 outcomes. Choosing 3 runners to advance to a final is a combination — the same three people in any order are one selection — so there are 8C3 = 336 ÷ 3! = 56.

The formulas: nPr = n! ÷ (n − r)! and nCr = n! ÷ (r! × (n − r)!). The combination is always the permutation divided by r!, the number of ways to order the r chosen items. That is why 10P3 = 720 but 10C3 = 120, and why a 5-card poker hand from 52 cards has 52C5 = 2,598,960 possibilities while dealing them in order gives 311,875,200.

With repetition. If items can be reused — digits in a PIN, scoops of ice cream from the same tub — the counts change. Ordered choices with repetition are simply nʳ: a 4-digit PIN from 10 digits has 10⁴ = 10,000 possibilities. Unordered choices with repetition (a "multiset") use the stars-and-bars result (n + r − 1)Cr: choosing 3 scoops from 6 flavours with repeats allowed gives 8C3 = 56 combinations.

Large numbers. Factorials explode: 20! is already over 2 × 10¹⁸. The calculator computes exactly with integer arithmetic up to 15 significant digits and switches to scientific notation beyond that, so 100C50 (≈ 1.009 × 10²⁹) still returns a correct value rather than overflowing.

Frequently asked questions

What is the difference between permutation and combination?
In a permutation the order of the chosen items matters (ABC and BCA are different); in a combination it does not (ABC and BCA are the same set). nPr is always larger than nCr for r > 1, by exactly a factor of r!.
What is 10 choose 3?
10C3 = 10! ÷ (3! × 7!) = 120. The matching permutation 10P3 = 10 × 9 × 8 = 720.
How many 5-card poker hands are there?
52C5 = 2,598,960, because a hand is a set of cards regardless of the order dealt. Enter n = 52, r = 5 with combinations selected, or press the preset.
When do I use repetition?
When the same item can be picked more than once: digits in a code, letters in a password, flavours in a multi-scoop order, dice rolls. Ordered with repetition is nʳ; unordered with repetition is (n + r − 1)Cr.
What if r is greater than n?
Without repetition it is impossible, so nPr and nCr are 0. With repetition it is fine — you can pick 5 items from 3 types if reuse is allowed (nʳ = 243 ordered, 7C5 = 21 unordered).
Why does the result show scientific notation?
Exact integers are shown up to 15 digits, which is the limit of double-precision arithmetic. Beyond that the calculator shows a correctly rounded value in scientific notation, e.g. 100C50 ≈ 1.00891 × 10²⁹.

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