Combinations & Permutations
What is the Combinations & Permutations?
Combinations and permutations both count the number of ways to select items from a group, but they differ in one crucial way: combinations don't care about order, permutations do. Choosing 3 pizza toppings from a menu of 10 is a combination โ the order you pick them in doesn't matter. Arranging 3 runners on a podium from a field of 10 is a permutation โ first, second, and third place are all different outcomes.
Because permutations count every possible ordering as distinct, P(n,r) is always at least as large as C(n,r) for the same n and r โ in fact, P(n,r) = C(n,r) ร r!, since each combination can be arranged in r! different orders.
What Each Variable Means
Units
| Quantity | Symbol | Unit |
|---|---|---|
| Combinations | C(n,r) | n! / (r!(n-r)!) |
| Permutations | P(n,r) | n! / (n-r)! |
When to Use It
- Combinations โ when selecting a group where the order doesn't matter (a committee, a hand of cards, toppings)
- Permutations โ when the order or arrangement matters (a ranking, a race finish, a PIN code)
- Probability problems that require counting the total number of possible outcomes
Step-by-Step Example
Problem: How many ways can you choose 3 people from a group of 5 (a) as an unordered group, and (b) as an ordered arrangement?
Use C(n,r) = n! / (r!(n-r)!).
C(5,3) = 5! / (3! ร 2!) = 120 / (6ร2) = 10Use P(n,r) = n! / (n-r)!.
P(5,3) = 5! / 2! = 120 / 2 = 60Interactive Calculator
Common Mistakes
Mistake: Using the permutation formula when order doesn't actually matter (or vice versa).
Fix: Always ask first: does rearranging the same items count as a different outcome? If yes, use permutations; if no, use combinations.
Mistake: Forgetting that 0! = 1, not 0.
Fix: By definition, 0! = 1 โ this matters when r = n, since (nโr)! becomes 0! in the formula.
Practice Questions
How many ways can you choose 2 books from a shelf of 6 (order doesn't matter)?
How many ways can 4 runners finish in 1st, 2nd, and 3rd place?
Hint: Order matters here, since the placements are distinct.
Frequently Asked Questions
How are combinations and permutations related?
P(n,r) = C(n,r) ร r! โ every combination of r items can be arranged in r! different orders, so permutations always count at least as many outcomes as combinations.
What if r is 0?
Both C(n,0) and P(n,0) equal 1 โ there's exactly one way to choose nothing at all.