Combinations Formula
Calculate the number of ways to choose items with C(n,r) = n! / (r!(n-r)!).
Learn when order does not matter in counting problems.
The Formula
The combinations formula counts the number of ways to choose r items from n total items.
Order does not matter — choosing ABC is the same as choosing CBA.
Variables
| Symbol | Meaning |
|---|---|
| C(n, r) | Number of combinations (also written as "n choose r") |
| n | Total number of items available |
| r | Number of items being chosen |
| n! | n factorial |
Example 1
A committee of 3 must be chosen from 8 people. How many ways?
n = 8, r = 3
C(8, 3) = 8! / (3! × (8 - 3)!) = 8! / (3! × 5!)
C(8, 3) = (8 × 7 × 6) / (3 × 2 × 1)
C(8, 3) = 336 / 6
C(8, 3) = 56 ways
Example 2
A lottery requires choosing 6 numbers from 49. How many possible tickets?
n = 49, r = 6
C(49, 6) = 49! / (6! × 43!)
C(49, 6) = (49 × 48 × 47 × 46 × 45 × 44) / (6 × 5 × 4 × 3 × 2 × 1)
C(49, 6) = 10,068,347,520 / 720
C(49, 6) = 13,983,816 possible tickets
When to Use It
Use the combinations formula when:
- The order of selection does not matter (teams, groups, committees)
- Choosing a subset from a larger set
- Calculating lottery or card game probabilities
- Working with the binomial theorem (binomial coefficients)