In a vernier callipers, (N + 1) divisions of vernier scale coincide with N divisions of main scale. If 1 MSD represents 0.1 mm, the vernier constant (in cm) is:

The correct option is:
$\frac{1}{100(N+1)}$

Step-by-Step Derivation:

1. Understand Vernier Constant (Least Count):
The vernier constant (VC), also known as the least count of the vernier calipers, is defined as the difference between the value of one main scale division (MSD) and one vernier scale division (VSD):
$\text{VC}=1\text{MSD}-1\text{VSD}$

2. Establish the relation between VSD and MSD:
According to the problem statement, $(N+1)$ divisions of the vernier scale coincide with $N$ divisions of the main scale. Therefore, we can write:
$(N+1)\text{VSD}=N\text{MSD}$
Solving for $1\text{VSD}$ gives:
$1\text{VSD}=\frac{N}{N+1}\text{MSD}$

3. Substitute 1 VSD back into the Vernier Constant formula:
$\text{VC}=1\text{MSD}-\frac{N}{N+1}\text{MSD}$
Taking the common factor of $1\text{MSD}$ out:
$\text{VC}=(1-\frac{N}{N+1})\text{MSD}$
Simplifying the expression inside the parentheses:
$\text{VC}=\left(\frac{N+1-N}{N+1}\right)\text{MSD}$
$\text{VC}=\frac{1}{N+1}\text{MSD}$

4. Convert the unit of 1 MSD to centimeters:
We are given that $1\text{MSD}=0.1\text{mm}$.
Since $1\text{cm}=10\text{mm}$, we convert millimeters to centimeters:
$1\text{MSD}=\frac{0.1}{10}\text{cm}=0.01\text{cm}=\frac{1}{100}\text{cm}$

5. Calculate the final Vernier Constant in centimeters:
Substitute the value of $1\text{MSD}$ in centimeters into the equation for VC:
$\text{VC}=\frac{1}{N+1}\times \frac{1}{100}\text{cm}$
$\text{VC}=\frac{1}{100(N+1)}\text{cm}$