= Solution
The literal $t\to qZ\gamma$ final state has a three-vertex <Loop diagram>[loop] using the <quartic W-photon-Z gauge coupling>[Standard Model quartic vertex] $W^+W^-Z\gamma$. The two other vertices are $t\to d_iW^+$ and $d_iW^+\to q$, with $q=u,c$. The internal <Down quark>[down-type quark] joins those two vertices, while two charged-vector propagators join them through the quartic vertex emitting both neutral bosons. The bottom panel of the figure shows this topology, satisfying the three-vertex hint without dropping the <photon>. Its additional electromagnetic coupling is contained in the four-leg vertex; counting vertices alone does not determine coupling order.
The companion single-$Z$ <Penguin diagram>[penguin] diagram illustrates $t\to qZ$: it instead attaches the neutral boson to the internal <quark> line and has three cubic vertices. These are different final states. A complete <Gauge invariance>[gauge-invariant] <Probability amplitude>[amplitude] requires all companion attachments; for $qZ\gamma$ these include <photon> radiation from charged lines as well as the quartic graph.
Both <Flavor eigenstate>[flavour] sums have the form
$$
\mathcal A(t\to qZ[\gamma])\propto\sum_{i=d,s,b}V_{ti}V_{qi}^*F_i,
$$
where the <Loop diagram>[loop] function also depends on the final-state <kinematics>. After cancellation of a mass-independent part by <GIM mechanism>, the <Bottom quark>[bottom-quark] <mass> gives the largest surviving contribution: $m_b^2\gg m_s^2,m_d^2$, while $|V_{tb}V_{cb}|\simeq0.042$ is not unusually small compared with the strange combination. The larger $|V_{cb}|$ than $|V_{ub}|$ then favours final <Charm quark>[charm]. Schematically,
$$
\boxed{t\to cZ\gamma\text{ with virtual }b\text{ is the favoured flavour combination},\qquad
\frac{|\mathcal A_c|^2}{|\mathcal A_u|^2}\sim\frac{|V_{cb}|^2}{|V_{ub}|^2}\simeq110.}
$$
The same <Flavor eigenstate>[flavour] preference applies to the separate two-body modes $t\to cZ$ and $t\to c\gamma$. This estimates a relative <Flavor eigenstate>[flavour] preference, not an absolute <branching ratio>. <Loop diagram>[Loop] factors and mass-difference suppression remain, so these channels are fully compatible with <GIM mechanism>[GIM]. A resolved extra <photon> also introduces another electromagnetic coupling and three-body <Lorentz-invariant phase space>[phase space]; an experimental $Z\gamma$ definition needs photon-energy and <Particle isolation>[isolation] cuts to distinguish hard radiation from the soft limit.
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