= Solution
Use fractional coordinates relative to conventional cubic vectors $(a\hat x,a\hat y,a\hat z)$. A <body-centered cubic lattice> has representative points
$$
(0,0,0),\quad(\tfrac12,\tfrac12,\tfrac12),
$$
so its conventional cell contains two <lattice points>: eight shared corners contribute one and the interior point contributes one. A <face-centered cubic lattice> has
$$
(0,0,0),\quad(0,\tfrac12,\tfrac12),\quad(\tfrac12,0,\tfrac12),\quad(\tfrac12,\tfrac12,0),
$$
so its conventional cell contains four: one from the shared corners and three from six half-shared faces.
For $N$ atoms at fractional positions $(x_j,y_j,z_j)$, the geometric <structure factor> and the atomic-factor-weighted amplitude are respectively
$$
G_{hkl}=\sum_{j=1}^N e^{2\pi i(hx_j+ky_j+lz_j)},\qquad
F_{hkl}=\sum_{j=1}^N f_j(\mathbf G)e^{2\pi i(hx_j+ky_j+lz_j)}.
$$
For identical atoms $F=fG$. Applying the conventional-cell lattice factors gives
$$
\boxed{G_{\rm BCC}=1+(-1)^{h+k+l}=\begin{cases}2,&h+k+l\text{ even},\\0,&h+k+l\text{ odd},\end{cases}}
$$
and
$$
\boxed{G_{\rm FCC}=1+(-1)^{k+l}+(-1)^{h+l}+(-1)^{h+k}
=\begin{cases}4,&h,k,l\text{ all even or all odd},\\0,&\text{mixed parity}.\end{cases}}
$$
These are lattice selection rules; an additional nontrivial <crystal basis structure factor> can extinguish further allowed reflections.
Of the eight equally represented parity triples, four satisfy the BCC condition and two the FCC condition. Thus the fractions among integer index triples are $1/2$ and $1/4$, respectively. This reflects the two and four equivalent <lattice points> in the conventional cells: their reciprocal lattices occupy corresponding subsets of the conventional reciprocal grids. In primitive reciprocal coordinates every integer triple denotes a reciprocal-lattice vector; lattice-centering absences are then built into the grid rather than an extra rule. A multi-atom chemical basis can still cause genuine additional absences. If permutations means merely reordering one fixed triple, its allowedness never changes: the quoted fractions concern all index triples, not reordering a chosen $(h,k,l)$.
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