$\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ are four towns. One can travel between $\mathrm{P}$ and $\mathrm{Q}$ along $3$ direct paths, between $\mathrm{Q}$ and $\mathrm{S}$ along $4$ direct paths, and between $\mathrm{P}$ and $\mathrm{R}$ along $4$ direct paths. There is no direct path between $\mathrm{P}$ and $\mathrm{S}$, while there are few direct paths between $\mathrm{Q}$ and $\mathrm{R}$, and between $\mathrm{R}$ and $\mathrm{S}$. One can travel from $\mathrm{P}$ to $\mathrm{S}$ either via $\mathrm{Q}$, or via $\mathrm{R}$, or via $\mathrm{Q}$ followed by $\mathrm{R}$, respectively, in exactly $62$ possible ways. One can also travel from $\mathrm{Q}$ to $\mathrm{R}$ either directly, or via $\mathrm{P}$, or via $\mathrm{S}$, in exactly $27$ possible ways. Then, the number of direct paths between $\mathrm{Q}$ and $\mathrm{R}$ is