= Solution
The given sphere has radius $a=5\,\mathrm{nm}$. The <Stokes–Einstein relation> at an illustrative $T=300\,\mathrm K$ gives
$$
D_A=\frac{k_BT}{6\pi\eta a}=4.39\times10^{-12}\,\mathrm{m^2\,s^{-1}}=4.39\,\mu\mathrm m^2\,\mathrm{s^{-1}}.
$$
For $N=30$ well-mixed independent searchers in cell volume $V$, the target-arrival rate is $Nk_a/V$. The corresponding mean first arrival time is
$$
\boxed{\tau_3\simeq\frac{V}{30\,4\pi D_Ar_0}
=0.121\,\mathrm s\left(\frac{V}{1\,\mu\mathrm m^3}\right)\left(\frac{5\,\mathrm{nm}}{r_0}\right)\left(\frac{300\,\mathrm K}{T}\right).}
$$
The source does not fix $V$, the reaction radius or <temperature>. For an order-of-magnitude estimate, a bacterial volume around $1\,\mu\mathrm m^3$ and reaction radius $5$–$10\,\mathrm{nm}$ give about $0.06$–$0.12\,\mathrm s$. A 20-base-pair B-form site is about $6.8\,\mathrm{nm}$ long, supporting a nanometre-scale capture radius but not uniquely defining its spherical value. The estimate is the ideal <diffusion>-limited time, not a measured intracellular search time: nonspecific binding, orientational matching, crowding and limited accessibility are excluded.
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