CombiLab

Permutation & Combination Calculator

How many ways can you arrange or choose r items from n?

Total items (n)

Items chosen (r)

Permutations (nPr)

720

nPr = n!/(n−r)! · nCr = n!/(r!(n−r)!)

Combinations (nCr): 120

What you entered

Total items (n): 10Items chosen (r): 3

1
n! (factorial of n)
10!= 3628800
2
(n − r)!
(10 − 3)!= 5040
3
Permutations: order matters
10! ÷ (10 − 3)!= 720
4
r! (factorial of r)
3!= 6
5
Combinations: order doesn't matter
10! ÷ (3! × (10 − 3)!)= 120

Result

Permutations (nPr): 720

Choosing 3 from 10 items: 720 ways if order matters, 120 ways if it doesn't.

Permutations and combinations of n items taken r at a time

Permutations count arrangements where order matters; combinations count selections where it does not. Permutations are always the larger of the two.

n	r	Permutations	Combinations	Ratio
5	2	20	10	2x
5	3	60	10	6x
10	2	90	45	2x
10	3	720	120	6x
10	5	30,240	252	120x
20	3	6,840	1,140	6x
52	5	311,875,200	2,598,960	120x

The ratio column is always r factorial, because every combination can be arranged in r! different orders - which is exactly the difference between the two counts. That is why the ratio is 2 when r is 2, 6 when r is 3, and 120 when r is 5, regardless of n. The last row is a five-card poker hand from a standard deck: 2,598,960 possible hands if order is irrelevant, which it is. Use permutations for passwords, race placings and seating; combinations for lottery tickets, committees and hands of cards.

Order matters, or it doesn't

Permutations count arrangements where order matters (1st, 2nd, 3rd place in a race), while combinations count selections where it doesn't (which 3 people are on a committee) — the same n and r produce a much larger permutation count.

Where these show up

Password/PIN possibilities, lottery odds, team selection, and probability problems in general all reduce to counting permutations or combinations correctly — getting the wrong one is one of the most common probability mistakes.

Frequently asked questions

A club of 20 people needs a 4-person committee. How many possibilities?

Order doesn't matter (just who is on it), so use combinations: C(20,4) = 20!/(4!×16!) = 4,845 possible committees. If they were choosing President, VP, Treasurer, Secretary instead, order matters: P(20,4) = 116,280.

How is this different from the permutations & combinations calculator?

Both calculate nPr and nCr. This version focuses on side-by-side comparison with explanation of when each applies. The other (permutations-combinations) shows the factorial breakdown in more detail. Same math, different presentation.

What are the odds of winning the lottery?

For a 6/49 lottery: C(49,6) = 13,983,816 possible combinations. Your chance with 1 ticket: 1 in ~14 million. That's roughly the same as flipping a fair coin and getting heads 23 times in a row.