In a row of (9) people, how many arrangements are possible if two particular people do not sit together?

Correct answer: (282240)

The direct answer is option A, 282240. There are 9 distinct people, so without any restriction the number of row arrangements is \(9! = 362880\). Let the two particular people be P and Q. Count the unwanted arrangements in which they sit together: treat P and Q as one block. Then there are 8 objects to arrange, giving \(8!\) arrangements, and P and Q can exchange places inside their block in \(2!\) ways. Thus unwanted arrangements are \(8!\times2! = 40320\times2 = 80640\). Therefore wanted arrangements are \(9! - 8!\times2! = 362880-80640=282240\). Option A is correct because it gives this difference. Option B, 362880, counts every arrangement and therefore includes arrangements where the two people sit together. Option C, 80640, is the number of arrangements where they do sit together, the opposite condition. Option D, 40320, is only \(8!\), so it forgets the two possible orders inside the block. Memory cue: for “not together”, calculate total minus “together”; for “together”, use one block and multiply by \(2!\).