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Given that, $(n-5)(n-10)-3(n-2) \leq 0$

$\Rightarrow n^{2}-10n-5n+50-3n+6 \leq 0$

$\Rightarrow n^{2}-18n+56 \leq 0$

$\Rightarrow n^{2}-14n-4n+56 \leq 0$

$\Rightarrow n(n-14)-4(n-14)\leq 0$

$\Rightarrow (n-4)(n-14)\leq 0$

$\Rightarrow 4\leq n \leq 14$

$\Rightarrow n \in [4,14]$

$\Rightarrow n=\{4,5,6,7,8,9,10,11,12,13,14\}$

$\therefore$ The total integers will be $11.$

Correct Answer $:\text{B}$

$\Rightarrow n^{2}-10n-5n+50-3n+6 \leq 0$

$\Rightarrow n^{2}-18n+56 \leq 0$

$\Rightarrow n^{2}-14n-4n+56 \leq 0$

$\Rightarrow n(n-14)-4(n-14)\leq 0$

$\Rightarrow (n-4)(n-14)\leq 0$

$\Rightarrow 4\leq n \leq 14$

$\Rightarrow n \in [4,14]$

$\Rightarrow n=\{4,5,6,7,8,9,10,11,12,13,14\}$

$\therefore$ The total integers will be $11.$

Correct Answer $:\text{B}$