= External-leg factorial cancellation at a quartic scalar vertex
{title2=$\mathcal M_{2\to2}^{\rm tree}=-\lambda$}
For a real <scalar field> with interaction $-\lambda\phi^4/4!$, the first <Dyson series> term is $-i\lambda\int\phi^4d^4x/4!$. Four labeled external legs can attach to the four fields in $4!$ ways, or in $\binom42\,2!\,2!=24$ ways after separating incoming and outgoing legs. The factorial cancels and spacetime integration supplies the <four-momentum conservation> delta function. This gives the connected tree <scattering amplitude> $-\lambda$. The separate <identical final-state symmetry factor> belongs in the event phase-space integral and is not another vertex factor.
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