3.6: 5-character passwords and a combinations equation

0606/12/F/M/23 — Question 7 · 6 marks

A 5-character password is to be formed from the following 13 characters.
Letters: A, B, C, D, E
Numbers: 9, 8, 7, 6, 5
Symbols: ∗*, #\#, !!
No character may be used more than once in any password.
(a)(i)
Find the number of possible passwords that can be formed.
[1]
(a)(ii)
Find the number of possible passwords that contain at least one symbol.
[2]
(b)
Given that (n12)×(n10)=(n+111)×(n−111)\dbinom{n}{12} \times \dbinom{n}{10} = \dbinom{n + 1}{11} \times \dbinom{n - 1}{11}, find the value of nn.
[3]