Derangements and Fixed Points
Count unchanged positions in a uniform random permutation and see why one missing match is impossible.
Free theory articleCount unchanged positions in a uniform random permutation and see why one missing match is impossible.
Concept & interpretation
A fixed point in a permutation is an element that remains in its original position. A derangement is a permutation with absolutely no fixed points. The number of derangements of n elements, denoted Dn, is the basis for determining the probability mass of any specific number of fixed points k in a uniform random permutation.
Experiment
To verify this in the Derangements and Fixed Points module:
- Set the Permutation size control to 4.
- Set the Fixed points probe control to 2.
- Observe the mass exactly matches 1/4 (0.25) in the scrollable count table.
- Notice the Derangements D(n) count is 9 and Permutations n! is 24.
- Change the Fixed points probe to 3, and confirm the mass is 0.