Permutation Calculator | nPr

A permutation is an ordered arrangement, choosing A then B is a different outcome from choosing B then A. A combination is an unordered selection, the group {A,B} and the group {B,A} are the same combination. The fundamental test: does the order of selection affect the outcome? If yes, count permutations P(n,r); if no, count combinations C(n,r).

• ›Passwords and PINs, order matters, "1234" ≠ "4321" → use permutations P(n,r)
• ›Lottery tickets, order doesn't matter, {1,7,23} is the same ticket however drawn → use combinations C(n,r)
• ›Race podium (1st, 2nd, 3rd), ordered positions → P(8,3) = 336 for 8 runners
• ›Committee of 5 from 20, unordered group → C(20,5) = 15,504
• ›Card hand in poker, unordered 5 cards from 52 → C(52,5) = 2,598,960

The relationship is always P(n,r) = r! × C(n,r). Each combination generates exactly r! permutations by rearranging the chosen items in all possible orders. For r = 1, P = C, a single item can only be arranged one way.

P(n,r) counts the number of ways to arrange r items chosen from n distinct items where the order of selection matters. The formula is P(n,r) = n! / (n−r)!, which simplifies to the product n × (n−1) × (n−2) × ··· × (n−r+1), exactly r consecutive integers multiplied together starting from n. Each step reflects one fewer choice as each position is filled.

• ›P(n,1) = n, trivially, n ways to pick one item
• ›P(n,n) = n!, all arrangements of n distinct items
• ›P(n,r) = 0 when r > n, cannot choose more items than exist
• ›P grows much faster than C: P(10,4) = 5040 vs C(10,4) = 210 (ratio = 4! = 24)

C(n,r), also written nCr, ⁿCᵣ, or "n choose r", counts the number of unordered selections of r items from n distinct items. The formula is C(n,r) = n! / (r! × (n−r)!). The r! in the denominator removes all possible orderings of the same group, converting the ordered permutation count into an unordered combination count. C(n,r) also appears as a coefficient in the binomial expansion (a+b)ⁿ.

• ›C(n,0) = C(n,n) = 1, only one way to choose nothing or everything
• ›C(n,1) = n, choosing one item from n
• ›C(n,r) = C(n,n−r), choosing r is the same count as leaving out r (symmetry)
• ›C(52,5) = 2,598,960, total distinct 5-card poker hands from a 52-card deck

When arranging n distinct objects around a circle, rotations of the same arrangement are considered identical, rotating everyone one seat to the right gives the same seating configuration. By fixing one object in place (eliminating the rotational equivalence), the remaining n−1 objects can be arranged in (n−1)! distinct ways. This is always equal to n!/n, the linear arrangements divided by the number of rotations.

• ›3 people: (3−1)! = 2 distinct seatings
• ›6 people at a dinner table: (6−1)! = 120 distinct seatings
• ›Necklace (flipping is also identical): (n−1)!/2, divide again by 2 for reflection symmetry
• ›5 beads on a bracelet: (5−1)!/2 = 12 distinct arrangements

P(n,r) = C(n,r) × r!, every unordered combination of r items generates exactly r! ordered permutations when the chosen items are rearranged in all possible sequences. Conversely, C(n,r) = P(n,r) / r!, dividing all ordered arrangements by the number of ways to reorder each group collapses them into unique unordered sets. This identity is foundational to the binomial theorem, Pascal's triangle, and probability theory.

• ›P ≥ C always; equal only when r = 0 or r = 1
• ›The ratio P/C = r! grows factorially, for r = 5, each combination yields 120 permutations
• ›Choosing a 5-person committee from 20: C(20,5) = 15,504 vs P(20,5) = 1,860,480 ordered selections

JavaScript's standard numbers are IEEE 754 64-bit doubles. They represent integers exactly only up to 2⁵³ ≈ 9 × 10¹⁵. Factorials grow much faster: 21! ≈ 5.1 × 10¹⁹ already exceeds this limit. A regular-number computation of 21! would silently round to the nearest representable double, losing the last few digits with no error or warning, giving a plausible-looking but wrong result. BigInt uses arbitrary-precision integer arithmetic, storing numbers exactly with no upper bound, at the cost of slightly slower arithmetic.

• ›20! = 2,432,902,008,176,640,000, fits in a 64-bit integer but NOT in a 53-bit double mantissa
• ›21! = 51,090,942,171,709,440,000, silent precision loss with regular Number
• ›P(200,100) has approximately 170 digits, completely impossible without BigInt
• ›C(200,100) ≈ 9 × 10⁵⁸, BigInt computes it exactly, Number gives garbage

The multinomial coefficient generalises C(n,r) to partitioning n objects into more than 2 groups simultaneously: n! / (k₁! × k₂! × ··· × kₘ!) where k₁ + k₂ + ··· + kₘ = n. It counts the number of distinct ways to divide n distinct objects into m ordered groups of specified sizes k₁, k₂, ..., kₘ. The binomial coefficient C(n,r) is the special case with m = 2 groups of sizes r and n−r.

• ›Arrangements of "MISSISSIPPI" (11 letters: 1 M, 4 I, 4 S, 2 P): 11!/(1!×4!×4!×2!) = 34,650
• ›Distributing 12 tasks equally among 3 teams of 4: 12!/(4!×4!×4!) = 34,650
• ›Multinomial theorem: coefficients in (x+y+z)ⁿ expansion come from n!/(k₁!k₂!k₃!)
• ›Card deal: 52 cards dealt 13 each to 4 players: 52!/(13!)⁴ ≈ 5.36×10²⁸ distributions