Combinations and Permutations Calculator

Order. A permutation is an ordered arrangement: A-B-C and C-B-A are two distinct permutations of {A, B, C}. A combination is an unordered selection: {A, B, C} is one combination no matter how you list it. The same n and r always give nPr greater than or equal to nCr, because every combination of size r can be re-ordered r! ways, so nPr = nCr * r!. Pick the mode in this calculator that matches your scenario: if shuffling the chosen items changes the answer (race rankings, PIN sequences, seat assignments), it is a permutation; if it does not (a hand of cards, a committee, a pizza topping selection), it is a combination.

Allow repetition when the same item can be picked more than once. A 4-digit PIN can be 0000 (the digit 0 used four times), so it is a permutation with repetition: 10^4 = 10,000 codes. A scoop-of-ice-cream selection where you can take two scoops of chocolate is a combination with repetition: with 3 flavors and 2 scoops, C(4, 2) = 6 multisets. Without repetition, each item is used at most once: nPr or nCr in the classical sense. The formulas differ: n^r and C(n+r-1, r) are the two with-repetition cases, n!/(n-r)! and n!/(r!(n-r)!) are the two without.

Imagine placing r distinct items into r ordered slots, drawing from a pool of n items without replacement. The first slot has n choices, the second has n-1 (one item is already used), the third has n-2, and so on, down to n-r+1 for the last slot. Multiplying these gives n * (n-1) * ... * (n-r+1), which is exactly n!/(n-r)! once you write out both factorials and cancel the (n-r)! tail. So the formula is just the product rule applied r times. The calculator uses the product form directly, not the factorial ratio, which keeps the intermediate values small and exact even when n is large.

Combinations dominate when grouping matters more than ordering. Poker uses C(52, 5) = 2,598,960 to count five-card hands. Lotteries use C(49, 6) = 13,983,816 for the odds of a six-from-49 jackpot. Sports drafts pick a team of k from a pool of n. Permutations show up when sequence matters. A password using 8 distinct lowercase letters has 26P8 = 62,990,928,000 arrangements. A 4-digit PIN with repetition allows 10^4 = 10,000 codes. Race-finishing orders, lineup orders, and lock-code sequences all use nPr or n^r.

JavaScript's standard number type is a 64-bit float that can store integers exactly only up to Number.MAX_SAFE_INTEGER, which is 2^53 - 1 = 9,007,199,254,740,991. Past that, every integer is rounded to the nearest representable value. Combinatorial counts blow past that limit fast: 21! already exceeds it, C(100, 50) is a 30-digit number. BigInt is JavaScript's arbitrary-precision integer type, with no upper bound short of memory. Using BigInt internally lets the calculator return the exact digits for any input up to n = 1000, then surface a scientific-notation summary when the value is too big to be useful as a number.