10 Choose 3: Permutations and Combinations
How many ways can you choose 3 of 10 items when order matters, and when it doesn't? Adjust the numbers below to try your own.
10 P 3 (order matters)
720
10 C 3 (order doesn't matter)
120
With 10 items, there are 120 ways to choose 3 of them when order doesn't matter (10C3), and 720 ways when order matters (10P3).
What is a Permutation and Combination Calculator?
Permutations and combinations both count the number of ways to select items from a larger set, but they answer subtly different questions. A permutation is a specific arrangement of elements where the order matters — swapping two chosen items produces a different permutation. A combination selects the same elements without regard for order — swapping two chosen items gives you the exact same combination.
This calculator computes both nPr (permutations of r items chosen from n) and nCr (combinations of r items chosen from n) at once, so you can see directly how much smaller the combination count is once order stops mattering and the "duplicate" orderings are folded together.
How nPr and nCr Grow as r Increases
With n fixed at 10, both counts grow rapidly as r increases toward n — but nPr always grows faster, since it keeps every distinct ordering that nCr discards. The chart uses a logarithmic scale because the gap between the two becomes enormous for larger r.
| r | nPr | nCr |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 10 | 10 |
| 2 | 90 | 45 |
| 3 | 720 | 120 |
| 4 | 5,040 | 210 |
| 5 | 30,240 | 252 |
| 6 | 151,200 | 210 |
| 7 | 604,800 | 120 |
| 8 | 1,814,400 | 45 |
| 9 | 3,628,800 | 10 |
| 10 | 3,628,800 | 1 |
The Formulas
Permutation (order matters): nPr = n! ÷ (n − r)!. For example, 11P2 = 11! ÷ 9! = 11 × 10 = 110 — there are 110 ways to arrange 2 items chosen from 11 when the two positions are distinct.
Combination (order doesn't matter): nCr = n! ÷ (r! × (n − r)!). Continuing the example, 11C2 = 11! ÷ (2! × 9!) = 55 — exactly half of 110, because every pair of items can be arranged in 2! = 2 orders, and combinations collapse those duplicate orderings into one.
A Real Example: Captain, Goalkeeper, and Strikers
Picture a soccer team choosing players for two different jobs. Choosing a captain and a goalkeeper from the squad is a permutation — captain and goalkeeper are different roles, so picking Alex as captain and Jordan as goalkeeper is a different outcome than picking Jordan as captain and Alex as goalkeeper. But choosing two players to be strikers is a combination — both strikers have the same role, so picking Alex-and-Jordan is identical to picking Jordan-and-Alex. Same two people, same two slots, but one scenario cares about order and the other doesn't.
Why Combinations Are Always Smaller
For any given r ≥ 2, nCr is always smaller than nPr for the same n and r, because every group of r items counted once by nCr corresponds to r! different orderings counted separately by nPr. That's exactly why nCr = nPr ÷ r! — dividing out the redundant orderings is what turns a permutation count into a combination count.
With Repetition (Not Covered Here)
This calculator handles selection without repetition — once an item is chosen, it can't be chosen again (like drawing lottery balls or dealing cards). Some problems allow the same item to be picked more than once (like choosing digits for a PIN code), which uses different formulas: permutations with repetition are simply nr, and combinations with repetition are (r + n − 1)! ÷ (r! × (n − 1)!). Both grow even faster than their without-repetition counterparts, since nothing is ever excluded from future picks.
Example — Your Current Inputs
With 10 items, there are 120 ways to choose 3 of them when order doesn't matter (10C3), and 720 ways when order matters (10P3).
Additional Example — A Lottery Draw
A 6/49 lottery draws 6 numbered balls from 49 without replacement, and the order they're drawn in doesn't matter for winning. That makes it a combination problem: 49C6 = 13,983,816 possible winning combinations. If the draw order mattered instead (say, matching balls to specific prize tiers by draw position), the count would jump to 49P6 = 10,068,347,520 — over 720 times larger, since 6! = 720 different orderings exist for every winning combination.
About These Parameters
- n (total items)
- The size of the full set you're selecting from — the number of people in a room, cards in a deck, balls in a lottery drum, or letters available for a code. Must be a whole number from 0 to 100.
- r (items chosen)
- How many of those n items are actually being selected or arranged. r can range from 0 (there's exactly one way to choose nothing) up to n (choosing every item at once). r can never exceed n — you can't select more items than exist in the set.
Frequently Asked Questions
How do I know whether a problem needs a permutation or a combination?
Ask whether swapping two of the selected items changes the outcome. If it does (like assigning first, second, and third place, or different job roles), use a permutation. If swapping them makes no difference (like picking a committee, a hand of cards, or a set of lottery numbers), use a combination.
What does nPr = nCr × r! mean in plain terms?
It means every unordered group counted by nCr can be rearranged in r! different ways, and each of those rearrangements is a separate permutation. So the permutation count is always the combination count multiplied by the number of ways to reorder each group.
What is 0! and why does it equal 1?
0! is defined as 1 by convention, not by multiplying anything. It makes the formulas consistent: nPn = n! ÷ (n − n)! = n! ÷ 0! must equal n! (there are n! ways to arrange all n items), which only works if 0! = 1.
Does this calculator support choosing with repetition allowed?
No — this tool covers selection without repetition (each item can be chosen at most once), which covers the large majority of real permutation and combination problems. The "With Repetition" section above gives the formulas for the repetition-allowed case if you need to work it out by hand.