Combinations
A combination is a selection in which order does not matter. Choosing r objects from n distinct objects can be done in ⁿCr ways.
Key formula
ⁿCr = n!/[r!(n − r)!]. Relationship with permutations: ⁿPr = ⁿCr × r!, so ⁿCr = ⁿPr/r!.
Useful properties
- ⁿC₀ = 1 and ⁿCn = 1
- ⁿCr = ⁿC(n−r) (symmetry)
Worked example
Worked example
In how many ways can a committee of 3 be chosen from 5 people? ⁵C₃ = 5!/(3!2!) = 120/(6×2) = 10. Order does not matter, so this counts unordered groups.
For a group with two categories, multiply the choices. Choosing 2 men from 4 and 1 woman from 3 gives ⁴C₂ × ³C₁ = 6 × 3 = 18 ways.
Remember
- Order does NOT matter in a combination.
- "Choose", "select" and "committee" signal combinations.
- Use ⁿCr = ⁿC(n−r) to simplify large r.