Quick Answer: If the order of the items matters, it is a PERMUTATION. If the order DOES NOT matter, it is a COMBINATION. A combination lock should actually be called a "permutation lock" because the order of the numbers matters (1-2-3 is not the same as 3-2-1).
Permutations: Order Matters
Think of a race with 10 runners. We need to award Gold, Silver, and Bronze medals. Does the order matter? Yes. Alice getting Gold and Bob getting Silver is entirely different from Bob getting Gold and Alice getting Silver.
Formula logic: There are 10 choices for 1st place. Once someone wins, there are 9 choices for 2nd place, and 8 for 3rd. So, 10 × 9 × 8 = 720 possible permutations.
Combinations: Order Doesn't Matter
Now imagine we are selecting 3 runners out of 10 to form a relay team. Does the order matter? No. A team of Alice, Bob, and Charlie is the exact same team as Charlie, Bob, and Alice.
Formula logic: Start with the permutations (720). But because order doesn't matter, we have counted the same team multiple times. How many ways can 3 people be arranged? 3 × 2 × 1 = 6 ways. So, divide the permutations by the redundant arrangements: 720 ÷ 6 = 120 combinations.
The Formulas (nPr and nCr)
- n: Total number of items
- r: Number of items being chosen
- Permutation (nPr): n! / (n - r)!
- Combination (nCr): n! / (r! × (n - r)!)
Lottery Example
When you play a 6/49 lottery, you pick 6 numbers from 49. The order they are drawn out of the machine does not matter. This is a combination. 49C6 = 13,983,816. Your odds of winning are 1 in 14 million.