Answer: (a) \(5040\), (b) \(720\), (c) \(3600\).

Answer: (a) \(5040\), (b) \(720\), (c) \(3600\).

(a) With no restrictions, the \(7\) students can be arranged in

\(7!=5040\)

ways.

(b) If the \(3\) boys are next to each other, treat them as one block. There are then \(5\) objects to arrange: the boys' block and the \(4\) girls.

The \(5\) objects can be arranged in \(5!\) ways, and the \(3\) boys can be arranged inside their block in \(3!\) ways. Hence the number of arrangements is

\(5!\times3!=120\times6=720.\)

(c) First count all arrangements, then subtract the arrangements where Cam and Dea are together.

If Cam and Dea are together, treat them as one block. There are \(6\) objects to arrange, and Cam and Dea can swap within their block in \(2!\) ways.

So the number with Cam and Dea together is

\(6!\times2!=720\times2=1440.\)

Therefore the number where Cam and Dea are not next to each other is

\(5040-1440=3600.\)