Combinations vs Permutations: When Order Matters

The one question that decides which to use

Before reaching for a formula, ask whether reordering the same items produces a new outcome. If it does, order matters and you need a permutation. If reordering changes nothing meaningful, order is irrelevant and you need a combination.

A quick test: a race podium is a permutation because gold, silver, and bronze are distinct positions. A handful of lottery numbers is a combination because the ticket wins regardless of the order in which the balls are drawn.

The two formulas side by side

The permutation formula is nPr = n! / (n − r)!, where n is the number of available items and r is how many you arrange. The combination formula divides that result by r! to cancel the orderings: nCr = n! / (r! · (n − r)!). The extra r! in the denominator is exactly why combinations are always the smaller count.

Note that n! (n factorial) means the product 1 × 2 × ... × n, and 0! is defined as 1. That convention keeps the formulas working when r equals n, where there is exactly one way to arrange or select everything.

A worked example with the same numbers

Take 5 people and pick 3. The permutations are nPr = 5! / (5 − 3)! = 120 / 2 = 60 ordered line-ups. The combinations are nCr = 5! / (3! · 2!) = 120 / (6 × 2) = 10 unordered groups. The ratio 60 / 10 = 6 is exactly 3!, confirming that each group of 3 can be ordered in 6 ways.

This relationship generalizes: nPr = nCr × r!. Whenever you already have a combination count, multiply by r! to get the permutation count, and divide by r! to go the other way.

Frequently asked questions

Is a lock combination really a combination?

No, it is mathematically a permutation. On a lock, 1-2-3 and 3-2-1 open different locks, so order matters and the correct term is permutation despite the everyday name.

Which is always larger for the same n and r?

Permutations are always at least as large, because nPr = nCr × r!. They are equal only when r is 0 or 1, since 0! and 1! both equal 1.

Does repetition change the formulas?

Yes. The formulas here assume each item is used at most once (no repetition). If items can repeat, you use different formulas, such as n^r for ordered selections with repetition.

How is nCr related to Pascal's triangle?

Each entry in Pascal's triangle is a combination count: the value in row n, position r, is nCr. That is why the triangle's numbers appear as the coefficients when you expand (a + b)^n.