$S = (1+11+111 + \cdots n \text{ terms})$
$=\dfrac 19 \times (9+99+999 + \cdots n \text{ terms)}$
$=\dfrac19 \Bigl ((\color{blue}{10}-\color{red}{1})+(\color{blue}{100}-\color{red}{1})+(\color{blue}{1000}-\color{red}{1}) + \cdots n \text{ terms} \Bigr )$
$= \dfrac19 \Bigl (\color{blue}{(10+100 + \cdots 10^n)} - \color{red}{(1+1+\cdots n \text{ terms)}} \Bigr )$
$=\dfrac 19 \left ( \color{blue}{\dfrac{10^{n+1} - 10}{10-1}} - \color{red}{n} \right )$
$=\dfrac 19 \left ( \dfrac{10^{n+1} - 10 - 9n}{9} \right )$
$=\dfrac{10^{n+1} - 9n - 10}{81}$
So, the correct answer is option A.