We have to find the number of letters mailed in $8^{th}$ set.

Let us identify the sequence in which the letters are mailed.

No. of letters mailed in $1^{st}$ set $=4$

Given that each one has to mail for 4 different persons.

$\therefore$ No. of letters mailed in $2^{nd}$ set $=4\times 4=16$

Similarly no. of letters mailed in $3^{rd}$ set $=16\times 4=64$

and so on.

$\therefore$ The sequence in this process is $4,16,64,........$

This series is a G.P. with first term $a=4$ and common ratio $r=4$

$\Rightarrow\:$ The no. of letters mailed in $8^{th}$ set $=t_8$ of the above sequence.

We know that $n^{th}$ term of a G.P. $=t_n=a.r^{n-1}$

$i.e.,$ The no. of letters in the $8^{th}$ set $=4\times 4^{8-1}=4^8=63504$

Given the postage charge for each letter =50 paise= $Rs.\large\frac{1}{2}$

$\therefore$ The postage charges for $8^{th}$ set of letters $=63504\times \large\frac{1}{2}$

$=