Option D is correct
Let $m=4(2^{2}) , n=2$ then $ s=100 \times 4 \times 2=800 \: numbers $
Let $S_1,S_2,S_3,S_4$ be subsets to $S,$ each having $(10 \times 2 \times 2=40) \: elements$.
Special condition is $ \mid S_i \cap S_j \mid \leq 2 \: (i.e \: \log m)$
Let
$ S_1= \{ 1,2, \ldots ,37,38,39,40 \} \\ S_2 = \{ 121,122, \ldots ,160 \} \\ S_3 = \{ 37,38,41,42, \dots ,78 \} \quad here \: \mid S_1 \cap S_2 \mid = 2 \\ S_4 = \{39,40,81,82, \ldots ,118 \} \quad \text{here also same}$.
and $A$ is an array of $800$ locations, initially $0$.
and $T \subseteq (S_1,S_2,S_3,S_4) \: and \: \mid T \mid = 2 \: \text{( i.e n)}$
$\text{Case 1: Let T } = (S_3, S_4)$
then $A$ is like this $$\begin{array}{c|cccccccccccccccc} & 1 & 2 & 3 & \ldots & 36 & 37 & 38 & 39 & 40 & 41 & 42 & \ldots & 78 & \ldots & 118 & \dots & 800 \\ \hline & 0 & 0 & 0 & \ldots & 0 & 1 & 1 & 1 & 1 & 1 & 1 & \dots & 1 & \ldots & 1 & \dots & 0 \end{array}$$ $Case \: \mathrm{a} \: $: $\begin{align}& \\ & \text{If input is $3$ (i.e. $S_3$) belongs to $T$ } \\ & \text{it went to one of the $40$ locations $\{ 37 \ldots 78 \}$} \\ & \text{cmp bit.} \\ & \text{$P$ ($S_3$ present) = $\dfrac{40}{40 }= 1$ } \end{align} $
$Case \: \mathrm{b} \: : \begin{align}& \\ & \text{If input is $1$ (i.e. $S_1$) belongs to $T$ } \\ & \text{it went to one of the $40$ locations $\{ 1 \ldots 40 \}$} \\ & \text{cmp bit} \\ & \text{$P$ ($S_1$ present) = $\dfrac{4}{40 }= \dfrac{1}{10} \: \mathsf{(wrong \: says)} $} \\ & \text{$P$ ($S_1$ not present) = $\dfrac{36}{40} = \dfrac{9}{10} = 0.9 \: \mathtt{(correct \: says)}$} \end{align}$