Combinations — A/L Combined Mathematics

A combination is an unordered selection of r distinct items chosen from a set of n distinct objects, denoted by ⁿC_r or (n \atop r) = n! / [r! (n - r)!]. While permutations count ordered arrangements (ⁿP_r = r! ⁿC_r), combinations count subsets where internal sequence is irrelevant. Key combinatorial principles include: (1) Partitioning sets into disjoint subgroups: selecting teams with specific gender/skill quotas using products of combinations (e.g., ᵐC_p × ⁿC_q); (2) Complementary Counting: finding combinations satisfying 'at least one' conditions by subtracting unwanted configurations from total unrestricted combinations: ⁿC_r(restricted) = Total - Unwanted; (3) Geometry applications: number of triangles formed by n points with no three collinear: ⁿC₃.

Check before you answer (A/L Pure Maths Paper 1)

1. Permutation vs Combination Selection: Choosing combinations (ⁿC_r) when order does not matter (committees, hands of cards, subsets) and permutations (ⁿP_r) for ordered arrangements.
2. Case Splitting Integrity: Dividing complex selection problems (e.g., 'at least 2 girls in a committee of 5') into disjoint mutually exclusive cases and summing them.
3. Complementary Counting Method: Applying Total - None = At least one to avoid lengthy multi-case summation.