Answer: (i) \(240240\), (ii) \(1800\), (iii) \(10080\).

There are \(5+6+3=14\) available characters.

(i) With no repetition and no restrictions, the number of possible codes is

\({}^{14}P_5=14\times13\times12\times11\times10=240240.\)

(ii) The first character must be one of the \(3\) symbols. The next two positions are filled by letters, and the final two by numbers. So the number of codes is

\(3\times {}^5P_2\times {}^6P_2 =3\times(5\times4)\times(6\times5) =1800.\)

(iii) The first two characters are numbers, so there are \({}^6P_2=6\times5\) ways to choose and arrange them.

The remaining three positions must be chosen from the \(5\) letters and \(3\) symbols, giving \(8\) non-number characters. Hence

\({}^6P_2\times{}^8P_3 =(6\times5)(8\times7\times6)=10080.\)