The ${\mathrm{S}}_{\mathrm{N}}1$ reaction rate depends on the stability of cabocation formed in the rate determining step.

$\underset{\mathrm{I}}{{\mathrm{CH}}_3{\mathrm{CH}}_2{\mathrm{CH}}_2{{\mathrm{CH}}_2}^{+}} \underset{\mathrm{II}}{{\mathrm{H}}_3{\mathrm{CCH}}_2{\mathrm{CH}}^{+}{\mathrm{CH}}_3} \underset{\mathrm{III}}{{\left({\mathrm{CH}}_3\right)}_2{{\mathrm{CHCH}}_2}^{+}} \underset{\mathrm{IV}}{{\left({\mathrm{CH}}_3\right)}_3{\mathrm{C}}^{+}}$

Tertiary carbocations are more stable than secondary and primary. Among the carbocations $\mathrm{I} \mathrm{and} \mathrm{III}$, $\mathrm{III}$ is readily formed than $\mathrm{I}$. Hence, the correct reactivity order is $\mathrm{I}<\mathrm{III}<\mathrm{II}<\mathrm{IV}$.