What's the difference between permutations and combinations?
The key question is whether order matters. A permutation is an arrangement where the order of selection is significant. A combination is a selection where only which items are chosen matters, not the sequence. Think of it this way: "ABC" and "BAC" are two different permutations of the same three letters, but they're the same combination.
Here's a quick test: if you're picking a 3-digit PIN, order matters (1-2-3 isn't the same as 3-2-1), so that's a permutation problem. If you're choosing 3 pizza toppings from a menu, order doesn't matter, so that's a combination problem.
How do you calculate P(n,r) permutations?
The permutation formula is P(n,r) = n! / (n-r)!, where n is the total number of items and r is how many you're arranging. The factorial n! means you multiply n by every positive integer below it down to 1. For example, P(5,2) = 5! / (5-2)! = 120 / 6 = 20. That means there are 20 ways to arrange 2 items from a set of 5 when order matters.
A common real-world example is a race with 5 runners. How many ways can the top 3 finish? That's P(5,3) = 5! / 2! = 120 / 2 = 60 different podium orderings.
How do you calculate C(n,r) combinations?
The combination formula is C(n,r) = n! / (r! × (n-r)!). It's the permutation formula divided by r! because in combinations you don't count different orderings of the same group as separate results. For example, C(5,2) = 5! / (2! × 3!) = 120 / 12 = 10. There are 10 ways to choose 2 items from 5 when order doesn't matter.
A classic combination problem: how many 5-card hands are possible from a standard 52-card deck? C(52,5) = 2,598,960. That's over 2.5 million distinct hands, which explains why poker's so unpredictable.
Permutations vs combinations reference table
For n = 10, here's how P(10,r) and C(10,r) compare as r increases:
| r | P(10,r) | C(10,r) | Ratio P/C = r! |
|---|---|---|---|
| 1 | 10 | 10 | 1 |
| 2 | 90 | 45 | 2 |
| 3 | 720 | 120 | 6 |
| 4 | 5,040 | 210 | 24 |
| 5 | 30,240 | 252 | 120 |
| 10 | 3,628,800 | 1 | 3,628,800 |
Notice how P(10,r) grows much faster than C(10,r) because you're also counting all the different orderings. And P(10,10) = C(10,10) × 10! = 1 × 3,628,800 = 3,628,800.
When is this used in real life?
Permutations and combinations show up constantly in probability, statistics, and everyday decisions. You'll use P(n,r) when order matters: ranking candidates, assigning finishing positions, generating passwords from a character set, or planning a scheduling sequence. You'll use C(n,r) when order doesn't matter: lottery ticket combinations, choosing a committee from a group, dealing card hands, or selecting ingredients for a recipe.
In probability, these formulas are the foundation for calculating the odds of events. If a lottery draws 6 balls from 49, the number of possible outcomes is C(49,6) = 13,983,816, which gives a 1-in-nearly-14-million chance per ticket.
Frequently asked questions
A permutation is an arrangement where order matters. P(n,r) = n! / (n-r)! counts the number of ways to arrange r items chosen from n distinct items in a specific order.
A combination is a selection where order doesn't matter. C(n,r) = n! / (r! × (n-r)!) counts the number of ways to choose r items from n items when the order is irrelevant.
Use permutations when order matters (passwords, race finishes, ranked lists). Use combinations when order doesn't matter (lottery picks, team selection, choosing toppings).
n is the total number of distinct items in the set. r is how many you're choosing or arranging. r must be less than or equal to n, and both must be non-negative whole numbers.
Factorial means multiply all positive integers down to 1. So 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow very fast: 20! is already over 2 quadrillion.
P(n,r) is always greater than or equal to C(n,r) because P(n,r) = C(n,r) × r! -- permutations count every different ordering of each combination, so there are r! times as many permutations as combinations for the same n and r.