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

$1$

$0$

$9$

$8$

$4$

$9$

$2$

$1$

$,$

$$your set would be

$$

$=$

$\{$

$$

$,$

I,

$$

$,$

$$

$,$

$$

$,$

$0$

$,$

$1$

$,$

$2$

$,$

$4$

$,$

$8$

$,$

$9$

$\}$

$.$

$($

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.