= Solution
The <weak charged current> spectator topology is $c\to qW^{+*}$, $W^{+*}\to u\bar q'$, with the initial $\bar u$ a spectator. The pair $q\bar u$ forms the negative <meson> and $u\bar q'$ the positive <meson>. The amplitude contains $V_{cq}^*V_{uq'}$ in one common charged-current convention; complex conjugating all factors changes no rate classification. With $c_C=\cos\theta_C$ and $s_C=\sin\theta_C$, the CKM magnitudes are
$$
\begin{array}{c|c|c}
\text{mode}&(q,q')&\text{amplitude order}\\\hline
\pi^+K^-&(s,d)&c_C^2\\
\pi^+\pi^-&(d,d)&s_Cc_C\\
K^+K^-&(s,s)&s_Cc_C\\
K^+\pi^-&(d,s)&s_C^2
\end{array}.
$$
Thus \b[$\pi^+K^-$ is favoured, both equal-flavour pairs are singly Cabibbo suppressed, and $K^+\pi^-$ is doubly suppressed]. The corresponding parametric rate hierarchy is $1:\tan^2\theta_C:\tan^4\theta_C$ after normalizing to the favoured mode. <Cabibbo suppression> counting alone does not rank the middle two: <phase space>, different matrix elements and <hadronic final-state interactions> matter. With identical constant matrix elements, the larger <pion> <phase space> would favour $\pi^+\pi^-$, whereas experimentally $K^+K^-$ is faster. The empirical decreasing order is
$$
\boxed{\Gamma(\pi^+K^-) > \Gamma(K^+K^-) > \Gamma(\pi^+\pi^-) > \Gamma(K^+\pi^-)}.
$$
As a later cross-check, <HFLAV>'s 2024 averages give approximately $4.00\%$, $0.411\%$, $0.149\%$ and $0.0138\%$ respectively. The exact middle ordering is an experimental input, not a consequence of the supplied masses and <Cabibbo angle> alone.<D0 hadronic branching fractions and Final State Radiation by HFLAV (2024)>[final branching-fraction average table and final doubly suppressed-mode estimate]{ref}
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2012/paper-1-charm-topology.png]
{title=Charm spectator topology}
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