= Solution
Treat the stated $120\,\mathrm{Mpc}$ as the radius of a complete, volume-limited search, with uniform galaxy <density> $n_g$. Its surveyed volume and number of galaxies are
$$
V=\frac{4\pi}{3}(120\,\mathrm{Mpc})^3=7.24\times10^6\,\mathrm{Mpc^3},\qquad
N_g=n_gV=7.24\times10^4.
$$
The observing time is $\Delta t=117/365.25=0.3203\,\mathrm{yr}$. One detection estimates
$$
\boxed{\nu_{\mathrm{BNS}}\simeq\frac1{N_g\Delta t}
=4.31\times10^{-5}\,\mathrm{yr^{-1}\,galaxy^{-1}},}
$$
or one merger per approximately $2.3\times10^4$ years per Milky-Way-type galaxy. The equivalent volume rate is about $431\,\mathrm{Gpc^{-3}\,yr^{-1}}$. The event's actual $40\,\mathrm{Mpc}$ distance is not the survey radius and must not replace $120\,\mathrm{Mpc}$ in the exposure.
A representative Galactic <core-collapse supernova> rate is about two per century, roughly $4.6\times10^2$ times larger than this merger-rate estimate. The historical $^{26}\mathrm{Al}$ estimate was $1.9\pm1.1$ per century.<Radioactive 26Al and massive stars in the Galaxy by Roland Diehl and others>[Abstract: Galactic core-collapse rate inferred from radioactive aluminium]{ref} The one-event <Poisson> uncertainty is large. Detector orientation, duty cycle, completeness and whether the quoted radius is an optimal horizon or an effective range also change the exposure; the boxed value is the specified simple-volume estimate.
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