Answer: (a) \(103680\). (b) \(270\).

(a) Treat the runners, swimmers and gymnasts as three blocks. The three blocks can be arranged in \(3!\) ways.

Within the blocks, the runners can be arranged in \(5!\) ways, the swimmers in \(4!\) ways and the gymnasts in \(3!\) ways.

Hence the number of arrangements is

\(3!\times5!\times4!\times3!=103680.\)

(b) Four members are selected, with at least one runner, one swimmer and one gymnast. Since there are three categories and four people, one category must contribute two people.

The possibilities are:

\({}^5C_2{}^4C_1{}^3C_1+{}^5C_1{}^4C_2{}^3C_1+{}^5C_1{}^4C_1{}^3C_2.\)

Therefore the number of ways is

\(10\cdot4\cdot3+5\cdot6\cdot3+5\cdot4\cdot3=120+90+60=270.\)