Permutations & Combinations

Read the formula sheet first, then work through the 10 questions below. Each answer comes with a step-by-step solution — read it even when you get the question right, the shortcut is often faster than your method.

Formula sheet

• ⁿPᵣ = n!/(n−r)! (order matters); ⁿCᵣ = n!/(r!(n−r)!) (order does not matter); ⁿCᵣ = ⁿCₙ₋ᵣ.
• Arranging n distinct things in a row = n!. In a circle = (n−1)!.
• Arranging n things with p alike, q alike = n!/(p! q!).
• Together condition: treat the group as one unit, then multiply by arrangements inside the group.
• Never together = Total − Together.
• Fundamental principle: AND → multiply, OR → add.
• Selecting r from n with a particular item always included: ⁿ⁻¹Cᵣ₋₁; always excluded: ⁿ⁻¹Cᵣ.

Practice

Pick an answer to see whether it is right and the worked solution. Your score is saved when you finish the set.