Given that,
- $f_{1}(x) = x^{2} + 11x + n$
- $f_{2}(x) = x$
And $f_{1}(x) = f_{2}(x)$
$\Rightarrow x^{2} + 11x + n = x$
$\Rightarrow x^{2} + 10x + n = 0$
If the equation has two distinct real roots, then
$b^{2}-4ac> 0$
$\Rightarrow 10^{2}-4(1)(n)> 0$
$\Rightarrow 100-4n> 0$
$\Rightarrow 100>4n$
$\Rightarrow 4n<100$
$\Rightarrow n<\frac{100}{4}$
$\Rightarrow n<25 $
$\Rightarrow\boxed{n_{\text{max}} = 24} $
$\therefore$ The largest positive integer value of $n$ is $24.$
Correct Answer $:\text{A}$