Comprehensive Calculator Guide
Combinations count the ways to choose r of n items when order does not matter and nothing repeats: C(n, r) = n! ÷ (r! × (n − r)!). A 3-person committee can be chosen from 10 people in 120 ways. Enter n and r for an exact answer, even for huge numbers.
From permutations to combinations
Ordering 3 of 10 people gives 10 × 9 × 8 = 720 permutations, but a committee is the same whatever the order of names, and each group of 3 appears 3! = 6 times. So 720 ÷ 6 = 120 committees: C(n, r) = P(n, r) ÷ r!.
Choosing r to keep is the same as choosing n − r to leave, so C(15, 11) = C(15, 4) = 1,365, and C(n, 0) = C(n, n) = 1. These numbers are the rows of Pascal's triangle.
Probability and repetition
If a committee of 3 is drawn at random from 10, any particular committee has a 1-in-120 chance. When items can repeat, such as 3 scoops from 5 flavours, use C(n + r − 1, r): C(7, 3) = 35, which you can compute here with n = 7 and r = 3.
How to Use
- 01Enter n, the total number of items.
- 02Enter r, how many to choose (0 ≤ r ≤ n).
- 03Read the exact count; use the permutation calculator if order matters.
Formula Used
C(n, r) = n! ÷ (r! × (n − r)!) = P(n, r) ÷ r!Practical Examples
Example 1: a committee
3 of 10 people: 120 committees.
Example 2: handshakes
12 people each shaking hands once: C(12, 2) = 66.
Example 3: a starting line-up
11 of 15 players: C(15, 11) = 1,365, the same as choosing 4 substitutes.
Important Tips
- Use the smaller of r and n − r when working by hand.
- Pairings such as handshakes or round-robin matches are always n × (n − 1) ÷ 2.
Common Mistakes to Avoid
- Using combinations when order matters, as with ranks or roles.
- Forgetting to divide by r! when converting from permutations.
Frequently Asked Questions
- How do I calculate combinations by hand?
- Compute the permutations and divide by r!: C(8, 3) = 336 ÷ 6 = 56.
- Why is C(10, 3) equal to C(10, 7)?
- Choosing 3 to take is the same as choosing 7 to leave.
- How many handshakes in a group?
- C(n, 2) = n × (n − 1) ÷ 2: 190 for twenty people.
- What about repetition?
- Use C(n + r − 1, r): choosing 3 from 5 kinds with repeats gives 35.