Answer: (a) \(8400\). (b)(i) \(3628800\). (b)(ii) \(1266\).

(a) To make a 5-digit number greater than \(50000\), the first digit can be \(5,6,7,8\) or \(9\). There are \(5\) choices.

The remaining four positions are filled by choosing and arranging four of the remaining eight digits:

\(5\times8\times7\times6\times5=8400.\)

(b)(i) Arrange the \(6\) people with brown hair first. This can be done in \(6!\) ways.

These \(6\) people create \(7\) gaps, including the two end gaps. To avoid adjacent red-haired people, the \(7\) red-haired people must occupy these \(7\) gaps, and they can be arranged in \(7!\) ways.

Therefore the number of arrangements is

\(6!\times7!=3628800.\)

(b)(ii) The number of ways to choose any \(5\) people is \({}^{13}C_5\).

The only choices not allowed are those with no brown-haired people, so all \(5\) chosen from the \(7\) red-haired people. There are \({}^{7}C_5\) such choices.

Hence the required number is

\({}^{13}C_5-{}^{7}C_5=1287-21=1266.\)