Six boys and four girls, all different, are to sit in a row of 10 chairs. In how many ways can they sit so that no two girls sit next to each other?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 604800
To keep a group apart, first seat the others, then put the group into the gaps between them, at most one in each gap.
The 6 boys sit in a row in \(6!=720\) ways
6 boys leave 7 gaps: one before the first boy, 5 between boys, one after the last boy
The 4 girls take 4 of the 7 gaps, one girl in each, in order: \(^7P_4=7\times6\times5\times4=840\) ways
Ways \(=720\times840=604800\)
Why the other options are incorrect
Option (1) 120960 — Incorrect. This is \(7!\times4!\), the count when the four girls sit together as one block.
Option (2) 25200 — Incorrect. This is \(6!\times\,^7C_4\), which chooses the 4 gaps but does not arrange the 4 different girls in them.
Option (3) 17280 — Incorrect. This is \(6!\times4!\), which arranges the boys and the girls but never chooses which gaps the girls take.