= Solution
There are \b[three vertex types], with all legs labeled in the following original drawings:
$$
\boxed{ggg,\qquad W^+W^-\gamma,\qquad W^+W^-Z.}
$$
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2022-gauge-cubic-vertices.png]
{title=The three fundamental cubic gauge-boson vertex types}
The <gluon> cubic <interaction vertex> follows from the non-Abelian <gauge field strength> of <Quantum chromodynamics>. The electroweak <Yang-Mills theory> <Lagrangian>, after expressing the fields in the physical charged and neutral <basis>, produces the two charged-<W boson> vertices. In particular there are no fundamental $\gamma\gamma\gamma$, $ZZZ$, $ZZ\gamma$, or $Z\gamma\gamma$ vertices, and the separate colour and electroweak gauge factors do not generate a mixed $gg\gamma$ or $ggZ$ vertex. The count is by the boson labels requested, not by separate assignments of <gluon> colour indices.
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