Combination Calculator
Calculate combinations C(n,r) = n!/(r!(n−r)!) for unordered selections.
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Quick Tip
Enter n, the total number of items to choose from.
About This Tool
Why use Combination Calculator?
A combination counts selections where order does not matter: C(n, r) = n! ÷ (r! × (n − r)!). Choosing 3 people from 8 gives 56 possible groups, because the same three people in a different order are still the same group.
The calculator also shows the permutation count and r!, which makes the relationship clear: combinations are permutations divided by the number of ways each selection could be ordered. A 6-from-49 lottery has C(49, 6) = 13,983,816 possible tickets, which is why a single ticket wins so rarely.
Key Benefits
Combination and permutation together
Seeing both, with r! as the divisor between them, makes it obvious which one a question needs.
Handles lottery-scale numbers
Large selections such as 6 from 49 are calculated directly instead of by listing outcomes.
Symmetry made visible
C(n, r) always equals C(n, n − r), a useful check: choosing 47 from 49 is the same count as leaving out 2.
How to use Combination Calculator
- 1
Enter n, the total number of items to choose from.
- 2
Enter r, how many you are selecting.
- 3
Read C(n, r), the number of distinct selections.
- 4
Compare with the permutation figure to see how much the order requirement would multiply the count.
Best use cases
Count how many teams or committees can be formed from a group.
Work out lottery odds for a given pick-and-pool format.
Count possible hands in a card game.
Answer combination questions in probability coursework.
Combination Calculator FAQs
What is the combination formula?
C(n, r) = n! ÷ (r! × (n − r)!). For C(8, 3): 40,320 ÷ (6 × 120) = 56.
When should I use combinations rather than permutations?
Use combinations when rearranging the chosen items does not create a new outcome — picking teammates, lottery numbers or pizza toppings. Use permutations for rankings, passwords and seating orders.
How do I calculate lottery odds?
Count the possible tickets with C(n, r) and take the reciprocal. For 6 numbers from 49 that is 1 in 13,983,816.
Why does C(n, r) equal C(n, n − r)?
Choosing which items are in is the same as choosing which are out. Picking 2 from 10 to include gives the same count as picking 8 to exclude: both are 45.
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