To solve this problem, we need to determine the time it takes for all three pipes running together to fill the cistern. Here's a step-by-step explanation:

Step 1: Understanding the Given Information

• We have three pipes of diameters 2 cm, 3 cm, and 4 cm.
• The smallest pipe, with a diameter of 2 cm, can fill the cistern in 232 minutes.
• The rate of water flow is proportional to the square of the diameter of the pipe.

Step 2: Calculating the Rates of Flow

• Let's assume the rate of flow of the 2 cm diameter pipe is proportional to \(2^2 = 4\).
• The rates of the other pipes will be proportional to their respective diameters squared:
  • The 3 cm pipe: \(3^2 = 9\)
  • The 4 cm pipe: \(4^2 = 16\)

Step 3: Calculating the Time Taken by All Pipes Together

• The 2 cm pipe fills the cistern in 232 minutes, so its work done in one minute is: \(\frac{1}{232}\) of the cistern.
• Since the flow rates are proportional to the squares of the diameters, we calculate the combined rate as the sum of individual rates:
  • Combined rate of flow per minute: \(\left(\frac{4}{4}\right) + \left(\frac{9}{4}\right) + \left(\frac{16}{4}\right) = 1 + \frac{9}{4} + \frac{16}{4} = 1 + 2.25 + 4\)
  • This simplifies to: \(7.25\)
• Thus, the three pipes together fill \(7.25\) times their rate of the smallest pipe.
• Total time taken by all pipes together = \(\frac{232}{7.25} = 32 \, \text{minutes}\).

Conclusion

• The time taken by all pipes running together to fill the cistern is 32 minutes.

This matches the correct answer: 32.