Write down the set $$ containing the letters of your first name and the digits in your

ECU student number. For example, if your name is JULIA and your student number

is $10984921,$ your set would be $=\{A,$I,$J,L,U,0,1,2,4,8,9\}.($Remember that

sets do not contain repetitions.$)$

Your set $$ should contain at least $3$ different letters and $4$ different numbers $-$ if it

does not then add in extra letters$/$numbers until this requirement is fulfilled.

$($a$)$ You are on the organising committee of ECU's new Maths Lovers Society and

need to create a website. Members of the society will need a password to

access the website, and you don't want hackers to guess the password.

You decide to create a password using the elements of the set $.$

Calculate the number of choices for each of the following password options,

and hence decide which format would be most secure$.$

$($i$)$ A password with $7$ characters, where the first three characters are

different letters from $$ and the last four characters are different

numbers from $.$

$($ii$)$ A password with $6$ characters, where the first four characters are

different elements from $,$ in a specific order, and the last two

characters can be any two different symbols from the set $\{$#$,$ $$,\%,$ &$,\backslash$

$>$ @$)$ in any order.

$($iii$)$ A password with either $5,6$ or $7$ characters, where the characters can

be any letters from $.$

$($b$)$ Design a format for an $8-$character password involving both the letters and

numbers in $.$ At least part of this format should include a permutation.

Solve the problem.