Counting and Combinatorial Proof Lab

Attempt P1–P5 after lessons 1–4. Define the counted set before computing, and save attempts before opening feedback.

1. P1: Count five-bit strings with exactly three ones. Give a bijection with subsets and enumerate one smaller instance to check the model.
2. P2: Count committees of four chosen from nine people with one chair. Count by committee-then-chair and by chair-then-members; prove the resulting identity.
3. P3: Split four distinct people into two groups of two, first with group names and then without. Contrast with binary necklaces of length four under rotation: list the rotation classes and explain why dividing by four fails.
4. P4: Count integers 1..100 divisible by 2, 3 or 5. Derive the signs by tracking an integer in exactly one, two or three forbidden-property sets.
5. P5: Count onto maps from three labeled inputs to two labeled outputs, then handle both empty-domain cases. Distinguish functions from partitions into unlabeled nonempty fibers.

Each response needs an exact integer and a no-overcount/no-undercount argument. A formula name alone does not demonstrate the counting model.