Combinations and Permutations
Reference for C(n,r) = n!/(r!(n-r)!) and P(n,r) = n!/(n-r)!.
Explains when order matters with worked examples for cards, teams, and lottery problems.
The Formulas
Permutations (order matters): P(n, r) = n! / (n - r)!
Combinations (order does not matter): C(n, r) = n! / (r!(n - r)!)
Combinations (order does not matter): C(n, r) = n! / (r!(n - r)!)
Permutations count arrangements where order matters (like ranking contestants). Combinations count selections where order does not matter (like choosing team members).
Variables
| Symbol | Meaning |
|---|---|
| n | Total number of items to choose from |
| r | Number of items being chosen |
| n! | n factorial = n × (n-1) × (n-2) × ... × 1 |
Example 1 — Permutation
How many ways can 3 runners finish 1st, 2nd, and 3rd out of 8?
Order matters (1st place ≠ 2nd place)
P(8, 3) = 8! / (8-3)! = 8! / 5! = 8 × 7 × 6
= 336 ways
Example 2 — Combination
How many ways can you choose 3 books from a shelf of 10?
Order does not matter (any group of 3 is the same)
C(10, 3) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1)
= 120 ways
Quick Decision Guide
Ask yourself: "Does the order of selection matter?"
- Yes, order matters → Use permutations (passwords, rankings, seat assignments)
- No, order does not matter → Use combinations (lottery, committees, card hands)
Key Notes
- The key distinction: Use permutations when order matters (e.g. podium finishes: 1st, 2nd, 3rd). Use combinations when order doesn't matter (e.g. selecting 3 people for a committee).
- Permutations formula: P(n,r) = n! / (n−r)!: The number of ways to arrange r items chosen from n, where arrangement order is significant.
- Combinations formula: C(n,r) = n! / (r!(n−r)!): Also written as "n choose r" or C(n,r). Each combination corresponds to r! permutations, so C(n,r) = P(n,r) / r!.
- Symmetry of combinations: C(n,r) = C(n, n−r). Choosing 3 items from 10 is the same count as choosing 7 to leave out. This symmetry is useful for mental checks.
- With repetition allowed: If items can repeat, permutations become nʳ and combinations become C(n+r−1, r). These are less commonly needed but important in probability theory.