Counting arrangements
A permutation is an ordered arrangement; a combination is a selection where order does not matter. The starting point is the factorial: n! = n × (n − 1) × … × 2 × 1, and by convention 0! = 1.
Key idea
nPr = n! / (n − r)! counts ordered arrangements of r objects from n. nCr = n! / (r!(n − r)!) counts unordered selections.
The link between them is nCr = nPr / r!, because each unordered selection can be ordered in r! ways. Use permutations when a first, second, third … matters (a race, a code); use combinations when only membership matters (a committee, a hand of cards).
Repeated objects
When some objects are identical, divide by the factorial of each repeat. The number of distinct arrangements of the letters in a word with repeats is n! divided by the product of the factorials of the repeated counts.
Worked example
A committee of 3 is chosen from 10 people. How many different committees are possible?
Order does not matter, so use combinations: 10C3 = 10!/(3! × 7!) = (10 × 9 × 8)/(3 × 2 × 1) = 720/6 = 120.
Remember
- Order matters → permutation; order does not → combination.
- Arrangements in a circle of n objects = (n − 1)!.
- nC0 = nCn = 1.