= Solution
The four FCC translations give a lattice <structure factor> that is nonzero only when $h,k,l$ are all even or all odd. The identical two-atom motif then multiplies it by
$$
1+\exp\left[\frac{i\pi}{2}(h+k+l)\right].
$$
For $(200)$ the FCC part is allowed, but the motif factor is $1+e^{i\pi}=0$. Hence
$$
\boxed{F_{200}=0.}
$$
In the projection, equal numbers of atoms occur at $x/a=0,1/4,1/2,3/4$. Their $(200)$ <phases> are respectively $+1,-1,+1,-1$. The paired layers cancel exactly for equal <silicon> <scattering amplitudes>. This <diamond-basis diffraction extinction> is additional to the FCC centring rule; it is not evidence that no atoms lie on the nominal reflecting planes.
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