nPr and nCr Relations: Formulas, Proofs & Examples

ⁿPᵣ counts ordered arrangements of r objects chosen from n, while ⁿCᵣ counts unordered selections. The two are linked by one clean relation: every unordered selection of r objects can be arranged in r! orders, so ⁿPᵣ = r! × ⁿCᵣ. This single equation connects the two halves of combinatorics.

This matters because it gives you a built-in consistency check: any permutation answer divided by r! must equal the combination answer. It also lets you compute whichever is easier and convert.

Key Concepts

• ⁿPᵣ: select r from n and arrange them — order matters.
• ⁿCᵣ: select r from n with no arrangement — order does not matter.
• r!: the number of ways to order any fixed set of r distinct objects.
• Two-step view: every permutation = a selection followed by an ordering.

Formulas

• Core relation: ⁿPᵣ = r! × ⁿCᵣ, valid for integers 0 ≤ r ≤ n.
• Solving for ⁿCᵣ: ⁿCᵣ = ⁿPᵣ / r! = n! / (r! (n − r)!).
• Solving for ⁿPᵣ: ⁿPᵣ = n! / (n − r)!.
• Pascal's rule (bonus relation): ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ, for 1 ≤ r ≤ n.

Worked Example

From 8 students, choose a team of 3 where the roles (captain, vice-captain, member) are distinct. First count unordered teams: ⁸C₃ = 8! / (3! 5!) = (8 × 7 × 6) / 6 = 56. Now order matters, so multiply by the 3! = 6 ways to assign roles within each team: ⁸P₃ = 56 × 6 = 336. Direct check: ⁸P₃ = 8 × 7 × 6 = 336. The two routes agree, confirming ⁿPᵣ = r! × ⁿCᵣ.

Common Misconceptions

• "ⁿPᵣ = ⁿCᵣ × n!." Wrong — the multiplier is r!, the factorial of the selected count, not of n.
• "The relation works for any r." It holds whenever 0 ≤ r ≤ n; beyond that range the symbols are not defined in this course.
• "Use ⁿPᵣ for committees." Wrong — committee membership has no order, so ⁿCᵣ is correct; using ⁿPᵣ overcounts by r!.

Real-World Applications

• Teams and committees (combinations) versus officer posts (permutations) from the same group.
• Lotteries: unordered draws use ⁿCᵣ, while ordered draws use ⁿPᵣ.
• Password analysis: converting between unordered character sets and ordered codes.
• Exam cross-checking: dividing a permutation answer by r! must reproduce the combination answer.

The bridge ⁿPᵣ = r! × ⁿCᵣ says: select, then order. Master this one relation and the whole permutation-combination chapter collapses into a single idea with a built-in way to verify every answer.