Probability
Permutations – Ordered Selections
A permutation counts the number of ways to choose and arrange r items from a set of n distinct items, where order matters. The formula is nPr = n × (n − 1) × … × (n − r + 1) – exactly r factors, counting down from n.
The key clue that a problem needs permutations, not combinations, is whether swapping the order of the chosen items creates a genuinely different outcome – like awarding 1st and 2nd place, where giving gold to Alex and silver to Sam is different from the reverse.
The Permutation Formula
nPr = n × (n − 1) × … × (n − r + 1), with exactly r factors. Equivalently, nPr = n! ÷ (n − r)!.
Evaluate 6P3.
6P3 = 6 × 5 × 4 = 120.
Real-Life Application
- Race results: counting the ways to award 1st, 2nd, and 3rd place.
- Passwords: counting ordered arrangements of a fixed set of characters.
- Scheduling: counting the ways to order a set of tasks or events.
Key Takeaways
- A permutation counts ordered selections, where order matters.
- nPr = n! ÷ (n − r)!.
- Use permutations whenever swapping the order changes the outcome.
Practice: Permutations
Permutations