We wish to make a microscope with the help of two positive lenses both with a focal length of $$20$$ mm each and the object is positioned $$25$$ mm from the objective lens. How far apart the lenses should be so that the final image is formed at infinity?

The given microscope uses two identical convex lenses — the objective and the eyepiece — each having focal length $$f_o = f_e = 20\text{ mm}$$.

Step 1: Position of the real (intermediate) image formed by the objective
For the objective, the object is placed $$u_o = 25\text{ mm}$$ in front of it (to the left). With the Cartesian sign convention, this is $$u_o = -25\text{ mm}$$. Using the thin-lens formula $$\frac{1}{v_o} - \frac{1}{u_o} = \frac{1}{f_o},$$ we get
$$\frac{1}{v_o} - \frac{1}{-25} = \frac{1}{20}$$
$$\frac{1}{v_o} + \frac{1}{25} = \frac{1}{20}$$
$$\frac{1}{v_o} = \frac{1}{20} - \frac{1}{25} = \frac{5 - 4}{100} = \frac{1}{100}.$$
Hence $$v_o = +100\text{ mm}.$$
The plus sign means the real, inverted intermediate image is formed 100 mm to the right of the objective.

Step 2: Condition for the final image to be at infinity
For the eyepiece to send the final rays parallel (image at infinity), its object (the intermediate image) must lie at its first focal point. Therefore the distance of this object from the eyepiece must equal the focal length of the eyepiece:
$$u_e = -f_e = -20\text{ mm}.$$
(The negative sign indicates the object is 20 mm to the left of the eyepiece.)

Step 3: Separation between the lenses
Let the distance between the objective and the eyepiece be $$L$$. The intermediate image is located 100 mm to the right of the objective and simultaneously 20 mm to the left of the eyepiece. Therefore
$$L = 100\text{ mm} + 20\text{ mm} = 120\text{ mm}.$$

Hence the two lenses must be kept 120 mm apart.

Option C which is: 120 mm