Permutations vs. Combinations: What is the Difference?
In mathematics, both permutations and combinations are used to count the number of possible outcomes in a given scenario. The critical difference between the two comes down to one simple rule: Does the order matter?
If the sequence or arrangement of the items matters, it is a Permutation. If you are just selecting a group of items and the order they are chosen in does not make a difference, it is a Combination.
When to Use Permutations (nPr)
Permutations are used for lists, passwords, seating arrangements, or race finishes.
Despite being called a "combination lock," a standard padlock actually requires a permutation. If your code is 1-2-3, entering 3-2-1 will not open the lock. The specific items (1, 2, and 3) must be entered in a strict sequence.
The Formula:
nPr = n! / (n - r)!
Where n is the total number of items, and r is the number of items being chosen.
When to Use Combinations (nCr)
Combinations are used for forming committees, mixing ingredients, or drawing lottery numbers.
If you buy a lottery ticket and choose the numbers 14, 28, 35, 42, and 50, it does not matter what order the machine draws those numbers in. As long as those five specific numbers are drawn, you win the jackpot. The order is irrelevant.
The Formula:
nCr = n! / [r! * (n - r)!]
Notice the extra r! in the denominator? This divides out the redundant arrangements, which is why combinations always result in a smaller number than permutations.
What does the Exclamation Point (!) mean?
In mathematics, the exclamation mark stands for a factorial. It means you must multiply a series of descending natural numbers. For example, 5! is calculated as 5 × 4 × 3 × 2 × 1 = 120. Because factorials grow incredibly fast, calculating permutations and combinations manually for large datasets is extremely difficult without our calculator.