Permutations vs Combinations: When Order Matters (and When It Does Not)
The difference between permutations and combinations confuses most students. One simple question resolves it every time.
Ask yourself: does the ORDER of selection matter? If rearranging the selected items creates a different outcome, use permutations. If rearranging does not matter, use combinations.
The Classic Examples
Picking a president, vice president, and secretary from 10 people: order matters (president Smith and VP Jones is different from president Jones and VP Smith). Use permutations: 10P3 = 720 ways. Picking 3 people for a committee from 10: order does not matter (the committee of Smith, Jones, and Lee is the same regardless of who was picked first). Use combinations: 10C3 = 120 ways.
Why Combinations Are Always Smaller
For every combination, there are r! permutations (all possible orderings of that group). 10C3 = 120, and each group of 3 can be ordered in 3! = 6 ways, so 10P3 = 120 × 6 = 720. Combinations collapse all orderings into one group, which is why there are fewer of them. If your answer for a combination problem is larger than the permutation answer, you mixed up the formulas.