1. Let one main scale division be $1\,\text{MSD} = m\,\text{units}.$

2. Given: $n\text{th division of main scale} = (n+1)\text{th division of vernier scale}.$ So, $n\,(\text{MSD}) = (n+1)\,(\text{VSD}).$

3. Substitute $1\,\text{MSD} = m$ units: $n m = (n+1)\,(\text{VSD}).$ Hence, $1\,\text{VSD} = \frac{nm}{n+1}.$

4. For a vernier caliper, least count is $\text{L.C.} = 1\,\text{MSD} - 1\,\text{VSD}.$

5. Therefore, $\text{L.C.} = m - \frac{nm}{n+1}.$

6. Simplify: $\text{L.C.} = \frac{m(n+1)-nm}{n+1} = \frac{m}{n+1}.$

7. So the least count is $\boxed{\frac{m}{n+1}}.$

8. Checking options:

   • A: $\frac{1}{n+1}$ — incorrect, missing factor $m$
   • B: $\frac{m}{n+1}$ — correct
   • C: $\frac{n}{n+1}$ — incorrect
   • D: $\frac{m}{n(n+1)}$ — incorrect

Hence, the correct option is B.