Combinations nCr
Choose r items from n
About This Calculator
A combination counts the number of ways to choose r items from a set of n items when the order of selection does not matter. Combinations are used in probability, statistics, and counting problems — from lottery odds to selecting a committee to calculating binomial coefficients.
Formula
C(n,r) = n! / (r! × (n−r)!) = nCr
Also written as: (n choose r) or the binomial coefficient
C(n,0) = C(n,n) = 1; C(n,1) = n
Example Calculation
How many ways to choose a committee of 3 from 7 people?
- n=7, r=3
- C(7,3) = 7! / (3! × 4!) = (7×6×5) / (3×2×1) = 210/6 = 35
There are 35 possible committees
Combination Values C(n,r)
| n \ r | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 5 | 1 | 5 | 10 | 10 | 5 | 1 |
| 6 | 1 | 6 | 15 | 20 | 15 | 6 |
| 7 | 1 | 7 | 21 | 35 | 35 | 21 |
| 8 | 1 | 8 | 28 | 56 | 70 | 56 |
| 10 | 1 | 10 | 45 | 120 | 210 | 252 |
Frequently Asked Questions
What is the difference between combinations and permutations?
Combinations C(n,r) count unordered selections — the group {A,B,C} is the same as {C,A,B}. Permutations P(n,r) count ordered arrangements — ABC is different from CAB. P(n,r) = C(n,r) × r!
Why is C(n,r) = C(n, n−r)?
Choosing r items to include is equivalent to choosing n−r items to exclude. These two operations always produce the same number of outcomes. For example, C(10,3) = C(10,7) = 120.
How do combinations relate to Pascal's triangle?
The values in Pascal's triangle are exactly the combination values C(n,r). Row n contains C(n,0), C(n,1), ..., C(n,n). This is why Pascal's triangle gives the coefficients in binomial expansions.
What is the birthday problem?
How many people must be in a room before there's a 50% chance two share a birthday? Using combinations and complementary probability, the answer is just 23 — surprising because each person is compared against all others, not just one.