$$5^x-5^{\left(x-3\right)}=124\cdot5^y$$
$$5^x\left(1-5^{-3}\right)=124\cdot5^y$$
$$5^x\left(1-\frac{1}{5^3}\right)=124\cdot5^y$$
$$5^x\left(1-\frac{1}{125}\right)=124\cdot5^y$$
$$\frac{5^x\left(125-1\right)}{125}=124\cdot5^y$$
$$5^x=5^3\cdot125$$
$$5^x=5^3\cdot5^3$$
$$5^x=5^{\left(y+3\right)}$$ $$x=y+3$$
$$x=y+3$$
$$y=x-3$$

$$ption\ is\ Option\ C$$