= Solution
For a disordered <face-centered cubic lattice>, equal average scattering factors on the four sites give reflections only when $h,k,l$ are all even or all odd. <Bragg's law> also requires $h^2+k^2+l^2\leq4a^2/\lambda^2$. With Fe K$\alpha$ <wavelength> $\lambda\simeq0.1936\,\mathrm{nm}$, the upper bound is $15.41$. The allowed sums are $3,4,8,11,12$, giving
$$
\boxed{111,\quad200,\quad220,\quad311,\quad222.}
$$
The next allowed family, $400$, has squared-index sum $16$ and cannot satisfy the diffraction condition for this <wavelength>. This list assumes the full geometrically accessible angular range, rather than an instrument-specific cutoff.
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