Let, the three consecutive positive integers be $x-1, x $ and $x+1.$
Now, $(x-1)(x )(x+1)=15600$
$\Rightarrow x(x^{2}-1^{2}) =15600$
$\Rightarrow x^{3}-x =15600$
$\Rightarrow x^{3}-x =(25)^{3}-25$
$\Rightarrow \boxed{x =25}$
So, we can get.
- $x-1=25-1=24$
- $ x+1=25+1=26$
$\therefore$ The sum of the squares of these integers $=24^{2}+25^{2}+26^{2}$
$\qquad \qquad = 576+625+676 = 1877.$
Correct Answer $:\text{D}$