= Solution
A <proportional hazards model> assigns individual $i$ the <hazard function> $h_i(t)=h_0(t)\phi_i$, where $h_0$ is the <baseline hazard> for multiplier one and $\phi_i>0$ is a time-constant <hazard multiplier>. In a <Cox proportional-hazards model>, $\phi_i=e^{x_i^T\beta}$ for a <covariate> vector $x_i$ and coefficient vector $\beta$; ratios $h_i(t)/h_k(t)=\phi_i/\phi_k$ are constant whenever the baseline is nonzero.
Assume <independent> individuals, <independent censoring> conditional on the <covariates>, and no tied events. Immediately before an event at $t$, condition on its observed <risk set> $R(t)$ and one event in a small time interval. The individual event <probability> is $h_0(t)\phi_i\,dt+o(dt)$, while the total is $h_0(t)\sum_{k\in R(t)}\phi_k\,dt+o(dt)$. Their limiting ratio is $\phi_i/\sum_{k\in R(t)}\phi_k$. Multiplication over observed events gives the <Cox partial likelihood>
$$
L_p(\beta)=\prod_{j:\text{event}}\frac{\phi_{i_j}}{\sum_{k\in R(t_j)}\phi_k}.
$$
The unspecified <baseline hazard> cancels. This is the usual successive conditional event contribution defining the <partial likelihood>, not the full event-time likelihood conditioned simultaneously on all event times.
For the four individuals the successive event <risk sets> are $\{a,b,c,d\}$, $\{c,d\}$ and $\{d\}$. Individual $b$ leaves at its <right censoring> time and contributes no event numerator. Consequently
$$
\boxed{L_{a,b,c,d}=\frac{\phi_a}{\phi_a+\phi_b+\phi_c+\phi_d}\frac{\phi_c}{\phi_c+\phi_d}\frac{\phi_d}{\phi_d}=\frac{\phi_a\phi_c}{(\phi_a+\phi_b+\phi_c+\phi_d)(\phi_c+\phi_d)}.}
$$
The last event contributes one and carries no further relative-risk information.
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