Answer: (a)(i) \(252\), (a)(ii) \(56\), (a)(iii) \(140\). (b) \(483840\).

Count the choices in stages, applying any restrictions before multiplying the number of possibilities together.

(a)(i) Choose 5 students from 10:

\({}^{10}C_5=252.\)

(a)(ii) If Abby and Ben are both included, choose the remaining 3 students from the other 8:

\({}^{8}C_3=56.\)

(a)(iii) Choose which one of Abby or Ben is included, then choose the other 4 students from the remaining 8 students:

\(2\times{}^8C_4=2\times70=140.\)

(b) If there are exactly three students between Abby and Ben, their positions must be 4 places apart. There are 6 possible pairs of positions, and Abby and Ben can be swapped in 2 ways.

The remaining 8 students can be arranged in \(8!\) ways.

So the number of arrangements is \(6\times2\times8!=483840\).

This gives the required answer: (a)(i) \(252\), (a)(ii) \(56\), (a)(iii) \(140\). (b) \(483840\).