Permutations, arrangements & combinations – choosing k from n items

MathematicsProbabilityAges 15–16

n items labeled A, B, C… sit in a row; students tap to pick k of them, either ordered or unordered, and compare against the full list of every possible choice. The count follows Pₙ = n!, Aₙᵏ = n!/(n − k)!, and Cₙᵏ = n!/(k!(n − k)!), each with its expanded product; Pascal's triangle below highlights the Cₙᵏ cell, showing that Cₙᵏ = Cₙ₋₁ᵏ⁻¹ + Cₙ₋₁ᵏ.