Given that, S = 4+44+444+... $$Rightarrow$$ $$S = frac{4}{9}(9+99+999+...)$$ $$Rightarrow$$ $$S = frac{4}{9}(10-1+10^2-1+10^3-1+...+10^n-1)$$ $$Rightarrow$$ $$S = frac{4}{9}(10+10^2+10^3+...+10^n-n)$$ $$Rightarrow$$ $$S = frac{4}{9}(frac{10(10^n - 1)}{10-1}-n)$$ $$Rightarrow$$ $$S = frac{40}{81}(10^n - 1) - frac{4n}{9}$$ Hence, option C is the correct answer. 
Alternate method:  Solving for n = 2 Sum of series = 4+44 = 48 Substituting n = 2 in options  (A) $$frac{40}{81}(8^{2}-1)-frac{5*2}{9}$$ = $$frac{280-10}{9}$$ = 30  (B) $$frac{40}{81}(8^{2}-1)-frac{4*2}{9}$$ = $$frac{280-8}{9}$$ =  $$frac{272}{9}$$
 (C) $$frac{40}{81}(10^{2}-1)-frac{4*2}{9}$$ = $$frac{440-8}{9}$$ = 48
 (D) $$frac{40}{81}(10^{2}-1)-frac{5*2}{9}$$ = $$frac{440-10}{9}$$ = $$frac{430}{9}$$