Pick a mode
Without repetition counts distinct items — committees, card hands, lottery draws. With repetition counts multisets where repeats are allowed, like 3 scoops from 12 flavours (double chocolate very much permitted).
01 · CALCULATOR
Calculate n choose r
Enter n and r, choose whether items may repeat, and get the exact answer with the full working — computed with big-integer math, so nothing ever gets rounded.
INPUT
C(n, r)
Every item is distinct and can be picked at most once — the classic nCr from textbooks.
TRY
C(5, 2)
10
STEP BY STEP
Runs entirely in your browser — nothing you type leaves this page. Whole numbers from 0 to 10,000 for both n and r.
02 · HOW TO USE
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02
03
Three steps to an answer
Enter n and r
n is how many items exist, r is how many you’re choosing. Whole numbers from 0 to 10,000. The engine is exact big-integer math — even C(10000, 5000) comes out correct, all ~3,000 digits of it.
Read — or take — the answer
The exact value leads, the full working follows, and a permutation comparison puts the number in context. One tap on the copy icon and “C(52, 5) = 2598960” is on its way wherever you need it.
03 · THE MATH
Combinations, in plain language
A combination counts the ways to choose r things out of n when order doesn’t matter. Grab 2 books off a shelf of 5 — Moby-Dick then Gilgamesh is the same pair as Gilgamesh then Moby-Dick, so it counts once. That shelf gives C(5, 2) = 10 pairs.
C(n, r) = n!r! × (n − r)!
- n! — every possible arrangement of all n items. We start by over-counting on purpose.
- r! — the orderings of the r chosen items that shouldn’t count separately.
- (n − r)! — the orderings of everything left behind, also over-counted.
Divide the over-count away and what survives is pure selection: the number of distinct r-item subsets. When repeats are allowed, the same bookkeeping done with stars and bars yields C′(n, r) = C(n + r − 1, r) — the mode this calculator switches to with one tap.
| Combination | Permutation | |
|---|---|---|
| Does order matter? | No | Yes |
| Formula | n!/(r!(n−r)!) | n!/(n−r)! |
| 5 items, choose 2 | 10 | 20 |
| Same 2 items | 1 selection | 2! = 2 orders |
| Feels like | a hand of cards | a finishing order |
04 · FAQ
Frequently asked questions
Short answers to the questions that come up most — the full working lives in the calculator above.
What is a combination?
A combination is a way of choosing r items from a set of n where order doesn’t matter. {A, B} and {B, A} count as one combination, not two. If you’re picking a team, it’s a combination; if you’re assigning roles inside the team, order kicks in and you’re in permutation territory.
What is the nCr formula?
C(n, r) = n! / (r! × (n − r)!). The numerator n! counts every arrangement of all n items — deliberate over-counting. Dividing by r! collapses the orderings inside your selection, dividing by (n − r)! collapses the orderings of what’s left, and what remains is pure choice. Worked example: C(5, 2) = 120 / (2 × 6) = 10.What’s the difference between a combination and a permutation?
Order. A permutation counts A-then-B and B-then-A as different outcomes; a combination doesn’t. That’s why
P(n, r) = C(n, r) × r! — every combination supports r! orderings — and why permutation counts are always the bigger (or equal) number. Choose 2 of 5: 10 combinations, 20 permutations.How do I calculate combinations with repetition?
When items can be chosen more than once, use the stars-and-bars formula:
C′(n, r) = C(n + r − 1, r). Three flavours, two scoops: C(4, 2) = 6 — chocolate-chocolate, chocolate-vanilla, chocolate-strawberry, vanilla-vanilla, vanilla-strawberry, strawberry-strawberry.What are n choose 0 and n choose n?
Both are exactly 1: there’s one way to choose nothing, and one way to take everything. Their cousins are just as tidy — C(n, 1) = n, and by symmetry
C(n, r) = C(n, n − r), since choosing r to keep is the same as choosing n − r to leave behind.What happens if r is greater than n?
Without repetition: 0 — you can’t pick more distinct items than exist. With repetition it’s perfectly legal, because items can be reused: C′(3, 5) = C(7, 5) = 21. The calculator tells you which situation you’re in.
How many 5-card poker hands are there in a 52-card deck?
C(52, 5) = 2,598,960 — every possible 5-card hand from a standard deck, which is why the 4 royal flushes feel like unicorns. Lotteries work the same way: Powerball’s white balls give C(69, 5) = 11,238,513 combinations, and your jackpot odds are 1 in that × 26.