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)
Key Notes
- Formula: C(n, k) = n! / [k!(n−k)!]: Read as "n choose k." Counts the number of ways to select k items from n without regard to order. Order does not matter — {A, B, C} and {C, A, B} are the same combination.
- Contrast with permutations: P(n, k) = n!/(n−k)! counts ordered arrangements. C(n, k) = P(n, k) / k! — dividing by k! removes the k! orderings that represent the same selection. "Choose a committee of 3" uses C; "assign President, VP, Secretary from 3" uses P.
- Symmetry: C(n, k) = C(n, n−k): Choosing 3 items from 10 is the same count as choosing which 7 to leave out. This symmetry halves the computation for large k (use k or n−k, whichever is smaller).
- Pascal's triangle: Each entry in Pascal's triangle is C(n, k). The identity C(n, k) = C(n−1, k−1) + C(n−1, k) shows how each row is built from the previous — adding adjacent entries gives the entry below and between them.
- Applications: Combinations calculate lottery odds C(49, 6) = 13,983,816, poker hand counts C(52, 5) = 2,598,960, committee selection, binomial probability coefficients, and the number of possible subsets of a set (total = 2ⁿ, since each element is either in or out).