Level 6: Counting with overlaps and stars

Counting and combinations · level 6 of 8 · rough age 13-14 (a guide for parents; anyone can start anywhere)

Level 5 gave you the choose numbers. This level gives you the tools that real counting problems need on top of them: fixing double counting when groups overlap, sharing out identical things with stars and bars, deciding which rule a problem needs, guaranteeing coincidences with the pigeonhole principle, and matching a hard collection with an easy one. It ends by using counting to measure chances, from lottery tickets to passwords.

Lesson	You will be able to
Inclusion-exclusion	Count the union of two or three overlapping sets, prove why the three-set rule counts everything once, and count things in none of the sets
Stars and bars	Share identical things among people, count whole-number solutions of \(a + b + c = n\), with minimums and maximums
Mixed problems: which rule?	Decide whether order matters, repeats are allowed and objects are alike, and pick the right one of the four basic counts
The generalised pigeonhole principle	Use \(\lceil n/k \rceil\) to guarantee coincidences, answer "how many to be sure", and prove the party problem
Counting by a bijection	Count by pairing one collection perfectly with another: subsets and strings, sums and gaps, crossings and sets of four points
Counting outcomes for probability	Find chances by counting equally likely outcomes, analyse a lottery fully, and explain why long random passwords are strong

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