How many passwords possible?

There are $11 choose 5$ ways to choose the places for the 5 $A$s. After this is done, there are $6$ spots remaining, and so there are $6 choose 3$ to choose the places for the 3 $Z$s.

There are now $11 - 5 - 3 = 3$ spots left. Now there are $24$ letters left to choose from (since $A$ and $Z$ have been used up), and the letters can be repeated, so there are $24^3$ ways to fill these $3$ spots.

Therefore the total is ${11 choose 5} cdot {6 choose 3} cdot 24^3 = 127 733 760 $.

First of all, note that eight of the letters are fixed. Now, the remaining three letters could be any of the letters except for $A$ and $Z$. There are ${}^{24}C_3$ combinations for this. Now, these 11 letters could interchange themselves in $11!$ ways.

But the interchanges among the $5$ $A$'s and $3$ $Z$'s are not significant. Hence, total number of permutations (interchanges) is $$frac{11!}{5!cdot 3!} = 332640$$

By the multiplication principle, total number of passwords is $${}^{24}C_3times 332640 = 4039518160$$

Hope it helps :)

Since $8$ places of the password are fixed, let's consider the remaining $3$ places which can be filled with the remaining $24$ letters.

The number of "$3-24$" passwords is $24^3$.

Now, consider a password composed by $5$ "A", $3$ "Z" and $3$ "*". The number of these passwords is $$frac{11!}{5!3!3!}.$$

Notice that this number is evaluated using the formula of permutations of multisets.

Therefore, the total number of passwords satisfying your requirements is:

$$24^3 cdot frac{11!}{5!3!3!} = 127'733'760.$$

Notice that the "*" letters corresponds to the "$3-24$" passwords.

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A more effective explanation is the following:

begin{array}{ccccccccccc} A & A & A & A & A & Z & Z & Z & * & * & * \ A & A & A & A & A & Z & Z & * & Z & * & * \ A & A & A & A & A & Z & * & Z & Z & * & * \ & & & & vdots \ * & * & * & Z & Z & Z & A & A & A & A & A \ end{array}

The number of rows of this table is $frac{11!}{5!3!3!}$. For each row, you can form exactly $24^3$ passwords assigning different letters to "*"s.