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.
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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.