Counting Principles

When two counted groups overlap, add the group sizes and then subtract the overlap once.

Use a permutation when order matters. Use a combination when order does not matter.

If you place more objects than boxes into the boxes, at least one box must contain more than one object. Use the ceiling idea for stronger versions.

Many counting errors come from not noticing whether choices happen in sequence or belong to separate non-overlapping cases.

Harder counting problems often become easier once you decide whether order matters, whether items repeat, and whether a restriction should be handled first or by subtraction.

Stars and bars counts ways to distribute identical objects into distinct boxes by turning the problem into positions for separators.