C

Step 1: Let the two positive charges be placed at points A and B, where charge 4e is at point A (0, 0) and charge e is at point B (λ, 0). We want to place a third charge q at point P, located at a distance x from charge 4e along the line connecting charges 4e and e.

Step 2: The forces acting on charge q due to the other two charges are given by Coulomb's law. The force exerted by charge 4e on charge q is:
$$ F_{4e} = k \frac{4e |q|}{x^2} $$ (toward the left if q is on the left of charge 4e)
The force exerted by charge e on charge q is:
$$ F_e = k \frac{e |q|}{(\lambda - x)^2} $$ (toward the right if q is on the right of charge 4e)

Step 3: For the charge q to be in equilibrium, the net force on it must be zero. Therefore, we set up the equilibrium condition:
$$ F_{4e} = F_e $$
Substituting the force expressions, we have:
$$ k \frac{4e |q|}{x^2} = k \frac{e |q|}{(\lambda - x)^2} $$
Dividing both sides by k |q| (assuming q is not zero), we get:
$$ \frac{4e}{x^2} = \frac{e}{(\lambda - x)^2} $$
Simplifying, we find:
$$ 4(\lambda - x)^2 = x^2 $$

Step 4: Taking the square root of both sides gives us:
$$ 2(\lambda - x) = x $$
This simplifies to:
$$ 2\lambda - 2x = x $$
Therefore: $$ 3x = 2\lambda $$, leading to $$ x = \frac{2\lambda}{3} $$
This means charge q should be placed at a distance of \frac{2\lambda}{3} from charge 4e to achieve equilibrium.

Step 5: Next, we need to analyze the stability of this equilibrium position. For the equilibrium to be stable, a small displacement in either direction must result in a force that pushes the charge back towards the equilibrium position.
- If q is displaced slightly toward charge 4e, the force due to 4e must increase (which happens since the distance decreases) and the force due to e must decrease (since the distance increases) to push it back. Thus, it is stable if $\frac{F_{4e}}{F_e}$ decreases.
- Conversely, if q is displaced toward charge e, the force due to e must increase to push it back toward the original position. Thus, it is unstable if $\frac{F_{4e}}{F_e}$ increases.

Thus, equilibrium is stable when charge q is between the two charges and unstable if placed outside. Therefore, the correct answer is option C.