Combinatorics
Core concepts
Understand the multiplication and addition principles, permutations vs combinations, circular arrangements, and the pigeonhole principle. Know when order matters (permutation) and when it does not (combination).
Traps
- Permutation vs combination: order matters → permutation; selection only → combination.
- "At least one" problems: use complementary counting (total minus none).
- Identical objects: divide by factorials of repeated items.
- Circular arrangements: fix one item, giving (n − 1)! arrangements.
- Overcounting when the same set is counted multiple times.
Rule of thumb
When a problem says "at least" or "not all," compute the total and subtract the complement.
Worked example
From 8 students, how many ways to choose a committee of 3? Order does not matter: C(8,3) = 8!/(3!·5!) = (8·7·6)/(3·2·1) = 56. If instead the three roles (chair, secretary, treasurer) are distinct, use P(8,3) = 8·7·6 = 336.