Repeated and circular arrangements

Repetition changes which orders are distinguishable

The letters in LEVEL have five positions, with two Ls and two Es. Temporarily labelling every copy gives 5! orders. Swapping the two L labels or the two E labels changes no visible word, so the distinct arrangements number 5!/(2!2!) = 30.

For repeated groups of sizes r₁,r₂,… summing to n, linear arrangements number n!/(r₁!r₂!…). Divide only for objects treated as indistinguishable. People sharing a first name remain distinct people; the experiment must specify which differences are ignored.

Restrictions still need their own counting structure. To keep identical letters apart, first arrange the other letters, then use gaps before, between and after them. Each visible arrangement should correspond to exactly one choice of the remaining-letter order and occupied gaps.

A circle needs an outcome convention

For n distinct people around an unlabelled circular table, suppose rotations represent the same arrangement and reflections remain different. Fix one person’s reference position, then order the remaining n-1 people clockwise. The count is (n-1)!.

Five distinct people therefore have 4! = 24 circular arrangements under that convention. Numbering the five seats instead gives 5! = 120 assignments, because rotating everyone now changes their seat numbers. If reflections as well as rotations are explicitly identified, the 24 clockwise orders pair into 12 mirror classes. For distinct objects this additional division by two applies when n ≥ 3.

Use a block only for the restriction it expresses

For six distinct people around an unlabelled table with rotations identified and reflections different, the unrestricted count is 5! = 120. If A and B must be adjacent, treat them as a block. Five circular units give 4! orders, and the block has two internal orders, giving 48.

A second method fixes A, puts B in either of A’s two neighbouring positions, then orders the remaining four people: 2 × 4! = 48. Under a uniform circular arrangement, adjacency therefore has probability 48/120 = 2/5, and non-adjacency has probability 3/5.

The gaps for a circle also differ from those for a line: the final gap leads back to the first object, rather than creating an extra end gap.

Repeated circular patterns can have different numbers of rotations

The distinct-object circular formula cannot simply be combined with every repetition correction. When a unique object exists, it can provide a reference: for circular BANANA patterns, fix the unique B and arrange A, A, A, N, N clockwise after it. This gives 5!/(3!2!) = 10 rotation classes, with reflections still distinguished unless they are already rotations of the same pattern.

With A, A, B, B there is no unique letter to anchor. List the six visible linear words and group rotations:

There are two rotation classes. Dividing the six linear words by four would give 1.5: the alternating pattern has fewer distinct rotations, so the proposed equal-size grouping is false. Direct classification resolves this small case without inventing a universal repeated-circle formula.

The experiment also changes the probabilities. If four distinct physical cards A₁, A₂, B₁ and B₂ are shuffled uniformly into four numbered positions, the six visible linear words are equally likely after erasing subscripts. Ignoring position numbers and identifying rotations then gives the AABB class probability 4/6 = 2/3 and the ABAB class probability 2/6 = 1/3. Choosing uniformly from the two circular pattern classes instead would give 1/2 each. Equal-looking diagrams do not specify the sampling mechanism.