Permutations, arrangements & combinations – choosing k from n items
MathematicsProbabilityAges 15–16
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Probability
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.
Lesson: Permutations, arrangements and combinations; Pascal's triangle (binomial theorem)
What it shows
Counting problems hinge on one question: does order matter? With n labelled items, a permutation arranges all of them (Pₙ = n!), an arrangement picks k of them in order (Aₙᵏ = n!/(n − k)!, often written ⁿPₖ), and a combination picks k without regard to order (Cₙᵏ = n!/(k!(n − k)!), often written ⁿCₖ). Listing every outcome makes the link visible: each combination of k items can be ordered in k! ways, so Aₙᵏ = k!·Cₙᵏ. Pascal's triangle holds the same numbers, each entry being the sum of the two above it.
How to use
Set Number of items n and Number chosen k, then pick a counting type with the Permutations Pₙ, Arrangements Aₙᵏ or Combinations Cₙᵏ buttons. Students tap items one at a time and watch the remaining choices shrink, which is the multiplication principle, then find their pick in the full list. Keep n and k fixed and switch between arrangements and combinations to compare the list lengths. Clear selection starts again; Show Pascal's triangle locates the Cₙᵏ cell.
Parameters you can change
- Number of items n 1–7
- Number chosen k 1–7
- Counting type Permutation (arrange all n items), Arrangement (choose k, ordered), Combination (choose k, unordered)
- Show Pascal's triangle
Questions to explore
- Choosing a captain, vice-captain and treasurer from 7 students: is this an arrangement or a combination, and how many ways are there?
- For n = 5 and k = 3, how many arrangements correspond to each combination, and why?
- Why is Cₙᵏ equal to Cₙⁿ⁻ᵏ, and where does Pascal's triangle show this symmetry?
