How this calculator works
An ordered selection distinguishes AB from BA. Without repetition, available choices decrease after each position; with repetition, they remain constant. Multiset mode divides the full factorial by the factorial of each identical-item count. Circular mode treats rotations as the same arrangement, but reflections remain different. Exact integer arithmetic preserves every digit. An empty ordered selection has one arrangement, including the conventional empty-string case 0^0 in repetition mode.
Formula
Without repetition: P(n,r) = n!/(n - r)!. With repetition: n^r. Multiset: N!/product(c_i!). Circular distinct items, rotations identified: (n - 1)!.
Worked example
Two ordered places chosen from five distinct items give 5 × 4 = 20 arrangements. A five-letter multiset with counts 2, 2 and 1 gives 5!/(2!2!1!) = 30 arrangements. Four distinct items around a circle give 3! = 6.
Advertisement
Frequently asked questions
How is a permutation different from a combination?
A permutation preserves order. A combination counts the same selected items once regardless of order.
Are mirror images the same in circular mode?
No. Only rotation is removed. Treating reflections as identical requires a different counting rule.
Why divide by factorials for repeated items?
Swapping indistinguishable copies does not make a new arrangement. Their internal permutations must be removed from the total.