Combinations

A combination is a selection where order does not matter. Choosing the team {Asha, Ravi} is the same as choosing {Ravi, Asha}; but making Asha captain and Ravi vice-captain is different from the other way round, which is a permutation.

Every selection of r objects can be arranged in r! orders, so ⁿPᵣ = ⁿCᵣ × r!. That gives

• ⁿCᵣ = n!/(r! (n – r)!), with ⁿC₀ = ⁿCₙ = 1.
• A committee of 3 from 7 people: ⁷C₃ = 7!/(3! 4!) = (7 × 6 × 5)/6 = 35.
• Handshakes among 8 people (each pair shakes once): ⁸C₂ = 28.
• Triangles from 10 points, no three in a line: ¹⁰C₃ = 120.

Choosing r is the same as leaving n – r: ⁿCᵣ = ⁿCₙ₋ᵣ. This makes big numbers easy: ¹⁰⁰C₉₈ = ¹⁰⁰C₂ = 100 × 99/2 = 4950.

It also solves equations. If ⁿC₇ = ⁿC₅, then either 7 = 5 (impossible) or 7 + 5 = n, so n = 12.

Pascal's rule: ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. Check: ⁵C₃ + ⁵C₂ = 10 + 10 = 20 = ⁶C₃.

Separate selections multiply. Choosing 2 men from 6 and 3 women from 5 for a committee: ⁶C₂ × ⁵C₃ = 15 × 10 = 150 ways.

Geometry example: a hexagon has 6 vertices. Joining any two gives ⁶C₂ = 15 lines, but 6 of them are sides, so it has 9 diagonals.