Permutations: Arrangements and Factorial Formula Guide

A permutation is an arrangement of objects in which the order matters. Choosing a president, secretary, and treasurer from 8 candidates is a permutation problem, because assigning the roles differently changes the outcome. The count is written ⁿPᵣ, read as "n permute r".

This matters because many real selections care about order: race positions, PIN codes, and seating rows all change when items are rearranged. Using permutations when order matters (and combinations when it does not) is the key decision in counting problems.

Key Concepts

• ⁿPᵣ: the number of ways to select and arrange r distinct objects out of n distinct objects.
• Order matters: the arrangement ABC is different from CBA in a permutation.
• Distinct objects: this formula assumes no two objects are identical.
• Range: n and r are non-negative integers with r ≤ n.

Formulas

• Permutation formula: ⁿPᵣ = n! / (n − r)!, for integers 0 ≤ r ≤ n.
• Expanded form: ⁿPᵣ = n × (n − 1) × … × (n − r + 1), a product of r factors.
• Full arrangement: ⁿPₙ = n!, the number of ways to arrange all n objects.
• Special values: ⁿP₁ = n and ⁿP₀ = 1.

Worked Example

In how many ways can 3 different prizes (first, second, third) be awarded to 8 students, with no student winning twice? Order matters because the prizes differ. Using the formula: ⁸P₃ = 8! / (8 − 3)! = 8! / 5! = (8 × 7 × 6 × 5!) / 5! = 8 × 7 × 6 = 336. Equivalently, the first prize has 8 possible winners, the second has 7 remaining, and the third has 6 remaining: 8 × 7 × 6 = 336.

Common Misconceptions

• "Use permutations when order does not matter." Wrong — that is a combination. Permutations are only for situations where rearranging changes the outcome.
• "ⁿPᵣ works when r > n." Wrong — you cannot arrange more objects than you have; r must be at most n.
• "Repeated objects use the same formula." Wrong — identical objects need the division formula n!/(p₁! p₂! …), not plain ⁿPᵣ.

Real-World Applications

• Race results and rankings: awarding distinct positions to contestants.
• PIN codes and passwords: ordered digit or letter sequences.
• Seating plans: assigning people to numbered chairs in a row.
• Sports: choosing a captain, vice-captain, and wicket-keeper from a squad.

Permutations count ordered arrangements through ⁿPᵣ = n!/(n − r)!. Whenever swapping two selected items creates a genuinely different outcome, reach for ⁿPᵣ — and when it does not, you need combinations instead.