Count the ways to choose r items from n when order does not matter: C(n, r), also written nCr or “n choose r”. Results are exact, even with hundreds of digits, and the factorial cancellation is shown.
C(10, 3)
120
Digits
3
Step-by-step working
FormulaC(n, r) = n! ÷ (r! · (n − r)!)
Cancel (n − r)! from the top, then divide by r! to remove orderingsC(10, 3) = (10 × 9 × 8) ÷ 3!
Result120
FormulaC(n, r) = n! ÷ (r! · (n − r)!) with repetition: C(n + r − 1, r)
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When order matters (and when it doesn’t)
Pick 3 of these 4 people for president, vice-president and secretary: that’s a permutation (order matters). There are 4 × 3 × 2 = 24 ways. But if you only pick a 3-person committee, every group of 3 appears 3! = 6 times in that list. So combinations = 24 ÷ 6 = 4.
Ana · Ben · Cai
Ana · Ben · Dee
Ana · Cai · Ben
Ana · Cai · Dee
Ana · Dee · Ben
Ana · Dee · Cai
Ben · Ana · Cai
Ben · Ana · Dee
Ben · Cai · Ana
Ben · Cai · Dee
Ben · Dee · Ana
Ben · Dee · Cai
Cai · Ana · Ben
Cai · Ana · Dee
Cai · Ben · Ana
Cai · Ben · Dee
Cai · Dee · Ana
Cai · Dee · Ben
Dee · Ana · Ben
Dee · Ana · Cai
Dee · Ben · Ana
Dee · Ben · Cai
Dee · Cai · Ana
Dee · Cai · Ben
The six highlighted rows are the same committee in different orders. Dividing by r! removes those duplicates: C(n, r) = P(n, r) ÷ r!.
Worked example
How many 5-card poker hands are there? C(52, 5) = 52 × 51 × 50 × 49 × 48 ÷ 5! = 311,875,200 ÷ 120 = 2,598,960. The (52 − 5)! on the bottom cancels with most of 52! on top, which is why you never need to compute 52! itself.
Lottery: choosing 6 numbers from 49 gives C(49, 6) = 13,983,816 possible tickets.
Combination or permutation?
Ask: does rearranging the same items give a different outcome? A committee of Ana, Ben and Cai is the same committee in any order: combination. President Ana, VP Ben and secretary Cai is different from President Ben…: permutation. There are always r! times more permutations than combinations.
Frequently asked questions
What is C(n, 0)?
1. There is exactly one way to choose nothing. Similarly C(n, n) = 1.
Why is C(n, r) = C(n, n − r)?
Choosing which r items to take is the same as choosing which n − r to leave behind.
What does “with repetition” mean?
Each item can be chosen more than once, like picking 3 scoops from 10 ice-cream flavours where two scoops can be the same flavour. The count is C(n + r − 1, r).