A row of seven with two friends together

Question

Seven students stand in a row for a photo. Two of them, Ana and Ben, insist on standing next to each other. How many arrangements of the row satisfy this?

Accepted Answer

Use the "block" trick. Tie Ana and Ben together into a single unit that moves as one. Now there are 6 units to arrange in the row: the Ana-Ben block plus the other 5 students. A row of 6 units can be ordered in 6! = 720 ways. But inside the block, Ana and Ben can stand in 2 orders (Ana-Ben or Ben-Ana), so multiply by 2: 6! 	imes 2 = 720 	imes 2 = 1440. Takeaway. To force two people adjacent, glue them into one block: arrange the (n-1) units as (n-1)!, then multiply by 2 for the block's internal order. Here 6! 	imes 2 = 1440.

A row of seven with two friends together

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