PickOrder

Permutations & Combinations

Permutations count ordered arrangements; combinations count unordered selections. Both formulas, both answers, every time.

n (total items)

r (items chosen)

Permutations P(10, 3)

720

Order matters

Combinations C(10, 3)

120

Order doesn't matter

Formulas

P(n, r) = n! / (n − r)!

C(n, r) = n! / (r! · (n − r)!)

Permutations and combinations of n items taken r at a time

Permutations count ordered arrangements; combinations count unordered selections. The ratio between them is always r factorial.

n	r	Permutations	Combinations
5	5	120	1
6	3	120	20
8	2	56	28
10	3	720	120
12	4	11880	495
20	2	380	190

The first row is the extreme case: choosing all 5 from 5 gives 120 orderings but only one possible selection, because there is exactly one way to pick everything. Rows one and two both give 120 permutations from different n and r, which is coincidence rather than pattern. The two-item rows are the easiest to sanity-check, since combinations are always exactly half the permutations when r is 2 - the two orders of any pair collapse into one selection. Use permutations for podium finishes and passwords, combinations for lottery draws and committees.

Order matters or not?

Use P(n,r) when arrangement matters — picking a 1st, 2nd, 3rd place. Use C(n,r) when it doesn't — picking a 5-person team from 10 people.

Frequently asked questions

What is the difference between permutation and combination?

Permutation: order matters. How many ways to arrange 3 books from 10? P(10,3) = 720. Combination: order doesn't matter. How many ways to choose 3 books from 10? C(10,3) = 120. The combination is always smaller because it doesn't count rearrangements of the same group.

How do I calculate nPr and nCr?

nPr = n! / (n−r)!. nCr = n! / (r! × (n−r)!). Example: n=10, r=3. P(10,3) = 10!/7! = 10×9×8 = 720. C(10,3) = 10!/(3!×7!) = 720/6 = 120. The key difference: nCr divides by r! to remove duplicate orderings.

When do I use permutations vs combinations in real life?

Permutations: PIN codes (1234 ≠ 4321), race finishing order, seating arrangements, license plates. Combinations: lottery numbers (order doesn't matter), team selection, choosing menu items, card hands in poker. Ask: 'Does rearranging the same items create a different outcome?' If yes → permutation. If no → combination.