How this instrument works
"Without repetition" means that once an item has been chosen, it is removed from the pool and cannot be chosen again for the same selection. Combined with order not mattering, that gives the standard combination formula: nCr = n! / (r! × (n − r)!). Because the pool never lets an item reappear, r can never exceed n here — you cannot draw more distinct cards from a deck than the deck contains.
It's worth naming the rule explicitly because a real alternative exists. If repetition were allowed instead — the same item could be selected more than once, like picking 3 ice cream scoops from 5 flavors where a flavor can repeat — the pool would never shrink as you choose, and the count would follow a different formula entirely: C(n + r − 1, r), the stars-and-bars formula, computed by this site's combinations-with-repetition instrument. The two diverge fast: choosing 5 of 20 without repetition gives C(20,5) = 15,504, while choosing 5 with repetition allowed from the same 20 types gives C(24,5) = 42,504 — noticeably more, because repeats open up outcomes the no-repeats rule forbids.
The classic example is a 5-card poker hand dealt from a standard 52-card deck. Each of the 52 cards can appear in a given hand at most once — a deck has exactly one ace of spades — so the count of distinct 5-card hands is C(52,5), a widely cited number in probability courses precisely because it's concrete, verifiable, and instantly recognizable to anyone who has counted poker odds.
- Enter the size of the full pool into n (total items) — for a standard card deck, that's 52.
- Enter how many items are being selected into r (items chosen, each used at most once) — 5 for a poker hand.
- Read nCr — combinations without repetition beneath the inputs; the instrument computes n! / (r! × (n − r)!) for you.
- If your scenario actually allows the same item to be chosen more than once, use the combinations-with-repetition instrument instead — the count follows C(n + r − 1, r), not n! / (r! × (n − r)!).
- If r exceeds n, the instrument flags it: without repetition, you cannot select more items than there are distinct items available to select from.
Worked example — 5-card poker hands from a 52-card deck
A standard deck has 52 distinct cards, and a poker hand draws 5 of them, with no card able to appear twice in the same hand and no distinction for the order they were dealt in. Enter n = 52 and r = 5. The instrument computes 52! / (5! × 47!), which works out to (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1) = 311,875,200 / 120 = 2,598,960. nCr reads 2,598,960.
That figure, 2,598,960, is the standard, widely cited count of distinct 5-card poker hands — every possible unordered combination of 5 cards from the 52-card deck, with no card repeated within a hand. If repetition were somehow allowed instead — the same card selectable more than once — the count would follow the stars-and-bars formula, C(52 + 5 − 1, 5) = C(56,5) = 3,819,816, a noticeably larger number, since repeats add outcomes that a real card deck's physical no-duplicates rule forbids.
Questions
What does "without repetition" mean here?
It means each item, once selected, is removed from the pool and cannot be chosen again for the same selection. Dealing the ace of spades into a poker hand means that exact card can't also appear a second time in the same hand — the pool of remaining cards shrinks by one with every card dealt, which is exactly what the n! / (r! × (n − r)!) formula assumes.
Why is C(52,5) = 2,598,960 the standard poker-hand count?
Because a poker hand is an unordered selection of 5 distinct cards from a 52-card deck, with no card able to repeat within a hand — precisely the combinations-without-repetition scenario. Plugging n = 52 and r = 5 into nCr = n! / (r! × (n − r)!) gives 2,598,960, a figure widely cited in probability courses because it's easy to verify and immediately recognizable.
What if repetition were allowed instead — how would the formula change?
The count would follow C(n + r − 1, r) instead of C(n, r) — the stars-and-bars formula this site's combinations-with-repetition instrument computes. For 5 cards from 52 types with repetition allowed, that would be C(56,5) = 3,819,816, noticeably more than 2,598,960, because repeats add outcomes a no-repeats deck can never produce.
Is this the same as the general Combination calculator on this site?
Yes — mathematically identical, nCr = n! / (r! × (n − r)!) either way. This instrument's labels and worked example spell out the "without repetition" assumption explicitly and contrast it directly against the with-repetition case, which is useful whenever the everyday word "combination" leaves it unclear whether repeats are allowed.
Does the order cards are dealt in change the poker-hand count?
No — a poker hand is defined by which 5 cards you hold, not the order they arrived in. If order did matter, the count would be a permutation instead, n! / (n − r)!, which for 52 cards taken 5 at a time gives 311,875,200 — exactly 5! = 120 times larger, since each unordered hand of 5 cards can be dealt in 120 different orders.
How is this different from a permutation without repetition?
Both assume each item is usable at most once, but a permutation without repetition counts order as significant while a combination without repetition does not. Dealing 5 specific cards to 5 named seats in order is a permutation (311,875,200 outcomes); dealing the same 5 cards into one undivided hand is a combination (2,598,960 outcomes).