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2 Counting configurations. A box with adiabatic walls contains a partition that di- vides its volume in the ratio 3:1. Initially the smaller region contains N molecules of an ideal gas at low density, and the larger is empty. A small hole is then made in the partition, so that the two regions come into equilibrium. Determine the number of configurations for the system as a function of the number of molecules in the smaller region, and hence find the most likely value for n. Com- pare the probability for this arrangement with that for a recurrence of the original configuration for the case N 100. Consider small fluctuations on about the most probable configuration. Using a Taylor expansion, or otherwise, find the distribution of on and show that its standard deviation is √3N/4. Hence, for one mole of gas, compute the fractional fluctuation in n.