Combination
Definition
A selection of objects where order does not matter. The number of combinations of n objects taken r at a time is C(n,r) = n! / (r!(n-r)!).
Why is Combination Important?
Combination is a foundational mathematical concept used across science, engineering, finance, and everyday problem-solving. From analyzing data sets to optimizing business decisions, this concept provides the analytical framework needed to interpret quantitative information accurately.
Our math calculators make complex computations simple and accessible, providing step-by-step results that help students, professionals, and curious minds explore mathematical relationships with confidence.
What is a Combination?
A combination is a selection of objects where order does not matter. Choosing players A, B, C for a team is the same as choosing C, A, B — both are the same combination.
Combination Formula
C(n, r) = n! / (r! × (n − r)!)
Also written as "n choose r" or ⁿCᵣ or (n r)
Examples
| Problem | Formula | Result |
|---|---|---|
| Choose 3 toppings from 8 | C(8,3) = 8!/(3!×5!) | 56 combinations |
| Lottery: 6 from 49 | C(49,6) | 13,983,816 (1 in 14 million odds) |
| Poker hand: 5 from 52 | C(52,5) | 2,598,960 hands |
| Team of 5 from 12 players | C(12,5) | 792 teams |
Pascal's Triangle and Combinations
Pascal's Triangle gives C(n,r) values. Each entry is the sum of the two entries above it:
- Row 0: 1
- Row 1: 1 1
- Row 2: 1 2 1
- Row 3: 1 3 3 1
- Row 4: 1 4 6 4 1
- Row 5: 1 5 10 10 5 1
Row n, position r gives C(n,r). Example: Row 5, position 2 = 10 = C(5,2).