A combination counts selections where order does not matter: C(n, r) = n! ÷ (r! × (n − r)!). Choosing 5 cards from a 52-card deck gives C(52, 5) = 2,598,960 hands. If items may repeat, the count becomes C(n + r − 1, r), so 3 scoops from 5 flavours gives 35.
Students working through probability homework, teachers setting exam questions, and anyone sizing a lottery draw or a team selection need the same number: how many distinct groups of r items can be drawn from n. This calculator returns that count exactly, digit for digit, using integer arithmetic so even C(500, 250) comes out without rounding. It also shows the number of ordered arrangements (permutations) for comparison, a scientific-notation form for very large answers, and the factorial expression behind the result. Switch to the repetition mode for multisets, where an item can be picked more than once. One limit: it counts selections from a single pool of distinct items; problems with identical items in the pool or several separate pools need the count built up by hand.
Choose No repetition for ordinary nCr, or With repetition when the same item can be chosen again.
Type the size of the pool you are choosing from, from 1 to 500.
Type how many items go into each group. Without repetition r cannot exceed n.
The exact number of combinations appears with the ordered count and the factorial working beneath it.
Result Number of combinations: 2,598,960
Ordered arrangements: 311,875,200
Scientific notation: 2.598 × 10^6
Digits in the result: 7
The number of distinct poker hands, where the order of the cards does not matter.
Result Number of combinations: 120
Ordered arrangements: 720
Scientific notation: 120
Digits in the result: 3
720 ordered line-ups collapse to 120 committees because each group of 3 appears in 3! = 6 orders.
Result Number of combinations: 35
Ordered arrangements: 125
Scientific notation: 35
Digits in the result: 2
Stars and bars turns this into C(7, 3), giving 35 different cups.
C(n, r) = n! ÷ (r! × (n − r)!); with repetition C(n + r − 1, r)
nr!Rather than computing three huge factorials, the tool multiplies (n − r + i) and divides by i for i = 1 to r, using exact big-integer arithmetic; every intermediate value is itself a binomial coefficient, so the division is always exact. It uses the smaller of r and n − r, since C(n, r) = C(n, n − r). For repetition, the stars-and-bars argument maps each multiset to a choice of r positions among n + r − 1 slots.
A permutation counts ordered arrangements, a combination counts unordered groups. Choosing 3 of 10 people gives 720 ordered line-ups but only 120 committees, because each committee of 3 appears in 3! = 6 different orders. The tool shows both numbers so you can see the factor between them.
Choosing 6 numbers from 49 without repetition gives C(49, 6) = 13,983,816 possible tickets, so a single ticket has roughly a 1 in 14 million chance of matching all six.
Choosing which r items to take is the same as choosing which n − r items to leave behind. Every selection of one kind corresponds to exactly one of the other, so C(10, 3) and C(10, 7) are both 120.
It means an item may appear in a selection more than once, as when buying 3 doughnuts from 5 varieties and taking two of the same kind. The count is C(n + r − 1, r), from the stars-and-bars method, and r may be larger than n in this mode.
Up to 500. The result is computed with exact whole-number arithmetic, so C(500, 250), which has about 150 digits, is shown in full along with a short scientific-notation form.
There is exactly one empty selection. The formula agrees: n! ÷ (0! × n!) = 1, because 0! is defined as 1 so that identities like this hold.
Using this tool in an assignment, article or report? Copy a citation, or a link for your website.
Tools Directory Editorial Team. (2026, October 26). Combination Calculator. Tools Directory. Retrieved October 26, 2026, from https://tools.milunir.com/math/algebra-tools/combination-calculator/
Tools Directory Editorial Team. “Combination Calculator.” Tools Directory, 26 Oct. 2026, https://tools.milunir.com/math/algebra-tools/combination-calculator/. Accessed 26 Oct. 2026.
Tools Directory Editorial Team. “Combination Calculator.” Tools Directory. Last modified October 26, 2026. Accessed October 26, 2026. https://tools.milunir.com/math/algebra-tools/combination-calculator/.
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