When order doesn't matter: When you just pick a group and the order inside doesn't matter — a team, a committee, a handful — use a selection, ⁿCᵣ.
See it: how many teams of 2 from 4 people?
Pick a team of 2 from Ann, Ben, Cara, Dan.
Step by step
- Pick 2 in order first: 4 then 3.
- But a team has no order — Ann-Ben is the SAME team as Ben-Ann, so every team is counted twice (2! = 2).
- So there are 6 teams: AB, AC, AD, BC, BD, CD.
Final answer
6 teams.
Pick 2 in order (4 × 3 = 12), then ÷ 2! because a team has no order → 6 teams.
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- how many you can choose from
- how many you choose (order ignored)
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ⁿCᵣ — the quick rule: Top: r numbers counting down from n. Bottom: r! = r × (r − 1) × … × 1.
Example: ⁵C₂ = (5 × 4)/(2 × 1) = 10.
IB-style question — a committee
A committee of 4 is chosen from 12 people.
How many different committees are possible?
Step by step
- Order doesn't matter → ¹²C₄. Quick rule: 4 numbers from 12 on top, 4! on the bottom.
- Work it out.
Final answer
495 committees.
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