One main scale division of a vernier caliper is equal to $$m$$ units. If $$n^{th}$$ division of main scale coincides with $$(n+1)^{th}$$ division of vernier scale, the least count of the vernier caliper is :

One main scale division (MSD) = $$m$$ units.

The $$n^{th}$$ division of the main scale coincides with the $$(n+1)^{th}$$ division of the vernier scale.

This means: $$n$$ MSD = $$(n+1)$$ VSD (vernier scale divisions).

Therefore: 1 VSD = $$\frac{n}{n+1}$$ MSD = $$\frac{nm}{n+1}$$ units.

Least count = 1 MSD - 1 VSD = $$m - \frac{nm}{n+1} = m\left(1 - \frac{n}{n+1}\right) = m \cdot \frac{1}{n+1} = \frac{m}{n+1}$$.

The correct answer is Option 3: $$\frac{m}{n+1}$$.