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Permutation Variants Calculator

Compute even, odd, and circular permutations from the size of a set.

Definitions

  • Even permutation — reachable from the identity using an even number of transpositions (swaps). Half of all permutations are even.
  • Odd permutation — reachable using an odd number of transpositions. The other half (when n ≥ 2).
  • Circular permutation — distinct arrangements around a circle, where rotations are considered the same. Equals (n − 1)!.

Permutation variants formulas

All permutationsn!
Even / odd permutationsn!2 each (n ≥ 2)
Circular arrangements(n − 1)!
Necklace / bracelet (flips count as the same)(n − 1)!2 (n ≥ 3)

Worked examples

  1. How many ways can 6 people sit around a round table?

    1. Fix one person to remove rotations: (6 − 1)! = 5!

    Answer: 120

  2. How many distinct bracelets from 6 different beads?

    1. 5!2 = 1202

    Answer: 60

  3. How many even permutations of 4 items?

    1. 4!2 = 242

    Answer: 12 (this is the group A₄)

Tips & common mistakes

  • A permutation is even if it can be made with an even number of swaps; 15-puzzle solvability depends on this.