Permutation vs. Combination: What's the Difference?

The core difference between permutations and combinations lies in a single question: Does order matter?

• Permutation: Order matters. A permutation is an arrangement of items in a specific order. Think of arranging books on a shelf, a batting order in baseball, or a password. `(a, b)` and `(b, a)` are two different permutations.
• Combination: Order does not matter. A combination is a selection of items where the order is irrelevant. Think of choosing a team of players, picking lottery numbers, or selecting toppings for a pizza. `{a, b}` and `{b, a}` are the same combination.

This calculator also handles scenarios with and without repetition (i.e., whether an item can be chosen more than once).

Formulas Used by the Calculator

1. Permutation without Repetition (nPr)

The number of ways to arrange 'r' items from a set of 'n' unique items.

nPr = n! / (n - r)!

2. Combination without Repetition (nCr)

The number of ways to choose 'r' items from a set of 'n' unique items, where order doesn't matter.

nCr = n! / (r! * (n - r)!)

3. Permutation with Repetition

The number of ways to arrange 'r' items from 'n' items where repetition is allowed.

n^r

4. Combination with Repetition

The number of ways to choose 'r' items from 'n' items where repetition is allowed and order doesn't matter. This is also known as the "stars and bars" method.

(n + r - 1)! / (r! * (n - 1)!)

The factorial calculations can result in very large numbers. For these, you might also find our Big Number Calculator useful.

Real-World Examples

• Lottery: Choosing 6 numbers from 49 is a combination problem because the order you pick them in doesn't matter. (Use nCr).
• Lock Combination: A 3-digit lock code is actually a permutation, because `1-2-3` is different from `3-2-1`. (Use Permutation with Repetition).
• Race Results: Finding the number of possible outcomes for 1st, 2nd, and 3rd place in a race with 10 runners is a permutation. (Use nPr).

These concepts are fundamental to many areas of mathematics, especially when working with our Probability Calculator.