There are different ways to approach this problem, but one possible method is to use Ohm's law and the rules for combining resistances in series and parallel. Here's one possible solution:

- First, recall that the resistance of a conductor is defined as the ratio of voltage to current, i.e. R = V/I. In other words, the resistance of a component is directly proportional to the voltage drop across it and inversely proportional to the current flowing through it. This means that if we know the voltage and current in a circuit, we can calculate the resistance of each component.
- Second, note that resistances in series add up, while resistances in parallel combine according to the formula 1/R = 1/R1 + 1/R2 + ... + 1/Rn, where R1, R2, ..., Rn are the individual resistances. In other words, in a series circuit, the total resistance is the sum of the individual resistances, while in a parallel circuit, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances.
- Now, let's apply these principles to the given problem. We have three resistances P, Q, and R, each of 2 ohms. We want to find the equivalent resistance of the circuit when these three resistances are connected in different ways.
- First, let's consider the case when the resistances are connected in series. This means that the current flows through each resistance in turn, and the voltage is divided among them. Since the resistances are the same, the voltage drop across each one is the same, and we can write Ohm's law as V = IR, where V is the total voltage, I is the current, and R is the total resistance. We also know that the voltage drop across each resistance is 2V/3 (since the total voltage is divided equally among them), so we can write V = 2V/3 + 2V/3 + 2V/3 = 6V/3 = 2V. Therefore, I = V/R = 2V/6 = V/3. Now we can use Ohm's law again to find the individual currents through each resistance: IP = VP/RP = (V/3) / 2 = V/6, IQ = VQ/RQ = (V/3) / 2 = V/6, IR = VR/RR = (V/3) / 2 = V/6. Finally, we can use the formula for resistances in series to find the total resistance: Rseries = RP + RQ + RR = 2 + 2 + 2 = 6 ohms. Therefore, the equivalent resistance of the circuit when the resistances are connected in series is 6 ohms.
- Second, let's consider the case when the resistances are connected in parallel. This means that the current is divided among the resistances, and the voltage drop across each one is the same. Since the resistances are the same, the current through each one is the same, and we can write Ohm's law as I = V/R. We also know that the total current is I = V/2, since the voltage drop across the parallel combination is the same as the total voltage. Therefore, the current through each resistance is I/3 = (V/2)