Consider dimer adsorption in one dimension. For $m \leq 3$, use conservation statements to find the following relations between $F_{m}$ and the empty interval probabilities $E_{m}:$
$$
\begin{aligned}
&F_{1}=1-E_{1} \\
&F_{2}=1-2 E_{1}+E_{2} \\
&F_{3}=1-3 E_{1}+2 E_{2}
\end{aligned}
$$
Notice that the general form of the last identity is
$$
F_{3}=1-3 \mathcal{P}[\circ]+2 \mathcal{P}[0 \circ]+\mathcal{P}[0 \times \circ]-\mathcal{P}[0 \circ \circ]
$$
but, for adsorption of dimers, $\mathcal{P}[0 \times \circ]=\mathcal{P}[\circ \circ \circ]$ so that the three-body terms in this identity cancel.
For $3<m<7$, express $F_{m}$ in terms of $E_{j}$ and the probability for two disconnected empty intervals $E_{i, j, k}$. For $F_{5}$, for example,
$$
F_{5}=1-5 E_{1}+4 E_{2}-2 E_{4}+2 P_{3}
$$