Ans: \((A)\)
We can manipulate the given equations as follows
$$
\begin{array}{ll}
\frac{1}{2} \mathrm{~B}+\frac{1}{2} \mathrm{C} \rightleftharpoons \mathrm{A}, & \mathrm{K}_{1}^{\prime}=1 \\
\frac{1}{2} \mathrm{C}+\frac{1}{2} \mathrm{D} \rightleftharpoons \mathrm{B}, & \mathrm{K}_{2}^{\prime}=\frac{1}{\sqrt{16}} \\
\mathrm{P} \rightleftharpoons \mathrm{C}+\frac{1}{2} \mathrm{D}, & \mathrm{K}_{3}^{\prime}=\frac{1}{\sqrt{25}}
\end{array}
$$
$$
\mathrm{P} \rightleftharpoons \mathrm{A}+\frac{1}{2} \mathrm{~B}, \quad \mathrm{~K}_{\text {final }}=1 \times \frac{1}{4} \times \frac{1}{5}=\frac{1}{20}
$$