Choose r items from n and see how many ways you can do it — ordered (permutations) or unordered (combinations), with and without repetition — plus the factorial working and Pascal’s triangle.
nPr = n! / (n − r)! = 5! / 2! = 60
nCr = n! / (r!·(n − r)!) = 10
A permutation counts arrangements where order matters: nPr = n!/(n − r)!. A combination counts selections where order does not matter, so it divides out the r! orderings of each group: nCr = n!/(r!·(n − r)!). nCr is therefore always nPr divided by r!.
When items can be picked more than once, permutations with repetition are simply nʳ, and combinations with repetition are C(n + r − 1, r). The nCr values are exactly the numbers in Pascal’s triangle: row n, position r.
Use a permutation when the order of the chosen items matters — a race podium, a PIN, a ranked list. Use a combination when only the group matters, not its order — a lottery draw, a committee, a hand of cards.
With repetition, the same item can be selected more than once — like a 4-digit code where digits can repeat. Without repetition each item is used at most once, like dealing distinct cards.
Every entry in Pascal’s triangle is a combination: the value in row n, position r is exactly nCr. Each number is the sum of the two above it, which mirrors the identity nCr = (n−1)C(r−1) + (n−1)Cr.
This calculator finds all four classical counting values at once: ordered selections without repetition (nPr), unordered selections without repetition (nCr), ordered selections with repetition (nʳ), and unordered selections with repetition. Enter the pool size n and the sample size r, and see every count instantly.
The headline figure is nCr — the number of ways to choose r items from n when order does not matter. Below it you also get nPr (order matters, no repeats), nʳ (order matters, repeats allowed), and the with-repetition combination count. The step-by-step breakdown shows the factorial expansion for nPr and nCr, and the Pascal's-triangle panel highlights exactly where your nCr value sits.
Choose 3 items from a pool of 5 (for example, selecting 3 toppings from a menu of 5).
nPr = 60 ordered arrangements; nCr = 10 unordered selections; with repetition: 125 ordered and 35 unordered.
A permutation counts arrangements where the order matters — ABC and BAC are two different outcomes. A combination counts selections where the order does not matter — ABC and BAC are the same outcome. In practice, use permutations for rankings, orderings, or sequences, and combinations for committees, teams, or subsets.
Use with-repetition counts when items can be chosen more than once — for example, picking digits for a password or selecting flavours when the same flavour can appear multiple times. The formula nʳ covers ordered choices with repeats; C(n+r−1, r) covers unordered choices with repeats.
170 is the largest integer whose factorial fits in a 64-bit floating-point number (170! ≈ 7.3 × 10³⁰⁶). Beyond that, JavaScript's Number type overflows to Infinity. The calculator uses the multiplicative formula rather than computing full factorials, so results are accurate up to the JavaScript safe-integer limit.
Permutations without repetition use nPr = n! / (n − r)!, computed multiplicatively as the product n × (n−1) × … × (n−r+1) to avoid overflow. Combinations without repetition use nCr = n! / (r!(n−r)!), computed with the symmetric multiplicative formula ∏(n−k+i)/i for i = 1..k where k = min(r, n−r). Ordered with repetition is simply nʳ. Unordered with repetition applies the multiset formula C(n+r−1, r), also computed multiplicatively.
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