First question: does order matter?: Before touching a formula, ask: does the order matter?
Arrange, row, rank, roles → order matters (ⁿPᵣ). Choose, select, team, committee → order doesn't (ⁿCᵣ). Then check for any restriction.
| When the question says… | It's… | Do this |
|---|---|---|
| arrange, in a row, rank, roles | an arrangement | n! or ⁿPᵣ |
| choose, select, team, committee | a selection | ⁿCᵣ |
| together, next to each other | a 'together' | glue into a block, × k! inside |
| not together, not adjacent | a separation | total − together, or the gap method |
| must include, always chosen | a required member | fix them, choose the rest |
| at least, at most | a group selection | add the cases, or total − unwanted |
| repeated letters | identical items | n! ÷ (factorial of each repeat) |
| shortest route on a grid | a grid route | choose which moves go 'up': ⁿCᵣ |
IB-style question — name it, then solve
A password is made from 3 different letters chosen from A, B, C, D, E, written in order.
How many passwords are possible?
Step by step
- Type? 'in order' and 'different' → an arrangement, no repeats.
- Choose and order 3 of the 5 letters.
Final answer
60 passwords.
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Same numbers, different type: Two questions can use the same numbers but need different tools. The type decides — so always name it first.
IB-style question — selection with a required member
A team of 3 is chosen from 7 players. The captain must be in the team.
In how many ways can the team be chosen?
Step by step
- Type? 'chosen' → a selection; 'captain must be in' → one place is fixed.
- Put the captain in, then choose the other 2 from the remaining 6.
Final answer
15 ways.
IB-style question — arrangement with a 'together'
6 books are arranged in a row on a shelf. Two particular books must be next to each other.
In how many ways can the books be arranged?
Step by step
- Type? 'arranged in a row' → an arrangement; 'next to each other' → a 'together'.
- Glue the 2 books into one block: 5 items arrange in 5!, and the block has 2 internal orders.
Final answer
240 ways.