The main scale of a vernier callipers has $\mathrm{n}$ divisions/cm. $\mathrm{n}$ divisions of the vernier scale coincide with $\left(\mathrm{n}-1\right)$ divisions of main scale. The least count of the vernier callipers is:

If ${\mathrm{n}}^{\mathrm{th}}$ division of vernier scale coincides with $\left(\mathrm{n}-1\right)$ divisions of main scale.

therefore, $\mathrm{n} \mathrm{VSD}=\left(\mathrm{n}-1\right) \mathrm{MSD}$

$\Rightarrow 1 \mathrm{VSD}=\frac{\left(\mathrm{n}-1\right)}{\mathrm{n}}\mathrm{MSD}$

$\therefore$   Least count $=1 \mathrm{MSD} - 1 \mathrm{VSD}$

$$\begin{gathered} =1 \mathrm{MSD}-\frac{\left(\mathrm{n}-1\right)}{\mathrm{n}}\mathrm{MSD}=1 \mathrm{MSD}-1 \mathrm{MSD}+\frac{1}{\mathrm{n}}\mathrm{MSD}=\frac{1}{\mathrm{n}}\mathrm{MSD} \\ =\frac{1}{\mathrm{n}}\times \frac{1}{\mathrm{n}}=\frac{1}{{\mathrm{n}}^2}\mathrm{cm} \left[\because 1 \mathrm{MSD}=\frac{1}{\mathrm{n}}\mathrm{cm}\right] \end{gathered}$$