Together → one block: When some people must stay together, glue them into a single block.
Arrange the blocks, then multiply by the ways to arrange them inside the block.
IB-style question — friends sitting together
5 friends sit in a row. Ana and Ben must sit next to each other.
In how many ways can the 5 sit?
Step by step
- Glue Ana + Ben into one block. Now arrange 4 things (the block + 3 friends).
- Inside the block, Ana and Ben can swap: 2 ways.
- Each of the 24 block-arrangements pairs with both internal orders, so multiply.
Final answer
48 ways.
The two who must sit together (here A and B) glue into one block: 4! × 2 = 48.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Start Studying Free Full access to Aimnova Pro · cancel anytime
A fixed internal order means × 1: If the order inside the block is fixed — "Ben immediately after Ana" — there's only 1 internal order.
So glue them as a block, but don't multiply by 2.
IB-style question — immediately after
8 runners finish a race with no ties.
In how many orders does Jack finish immediately after Andrea?
Step by step
- 'Jack immediately after Andrea' glues them as a block — and the order inside is fixed (Andrea, then Jack).
- Arrange the block plus the other 6 runners = 7 things.
- Internal order is fixed, so × 1.
Final answer
5040 orders.