5.3.1—Permutations and combinations

For repeated identical items, divide by factorials for duplicate arrangements; for selections with repetition, distinguish stars-and-bars style counting from ordinary combinations.

Write a small case first to check what an outcome means, then generalise. Position restrictions can change the formula entirely.

The distinct arrangements of A,A,B,C are 4!/2!, because swapping the A’s creates no new arrangement.

Treating identical objects as labelled creates artificial outcomes and inflates the count.