5.2 Permutations and combinations

nPr counts ordered arrangements of r objects from n; nCr counts unordered selections. Use nPr=n!/(n−r)! and nCr=n!/[r!(n−r)!].

Decide whether positions matter before choosing a formula, and avoid counting the same outcome under different descriptions.

Choosing president and secretary from 8 people uses 8P2; choosing a two-person committee uses 8C2.

The same two people can form two ordered roles but only one unordered committee.

For arrangements with restrictions, count valid positions directly, treat linked objects as a block, or subtract forbidden arrangements from the total.

State whether objects are distinct, whether rotations are identified, and whether repetition is allowed before counting.

Arranging 5 books with two specified books together gives 4!×2!, treating the pair as one block and ordering the pair internally.

A block has internal arrangements; forgetting them undercounts, while counting them twice overcounts.