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 $=63504\times \large\frac{1}{2}$
$=