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Chapter 20: Problem 7

The number of macrostates that can result from rolling a set of \(N\) six-sided dice is the number of different totals that can be obtained by adding the pips on the \(N\) faces that end up on top. The number of macrostates is a) \(6^{N}\) b) \(6 N\) c) \(6 N-1\). d) \(5 N+1\).

Short Answer

Expert verified
Answer: d) \(5N+1\)

Step by step solution

01

Option a: \(6^{N}\)

This option implies that there are \(6\) possible outcomes for each die, and since we have \(N\) dice, we would have \(6^{N}\) different combinations. However, this does not represent the correct number of macrostates, as this would instead show the total number of different combinations without considering the totals of the pips on top.
02

Option b: \(6 N\)

This option suggests that the number of macrostates is linearly dependent on the number of dice. This is not correct, as each die can contribute different totals (from 1 to 6), and the number of possibilities should generally increase with having more dice.
03

Option c: \(6 N-1\)

Similar to option b, this option also suggests that the number of macrostates is linearly dependent on the number of dice. It is still incorrect because it does not represent the increase in the number of possibilities as the number of dice increases.
04

Option d: \(5 N+1\)

For 1 die, the minimum and maximum values that can be obtained are 1 and 6, which means there are 6 different totals (macrostates) possible. Now, let's assume we have 2 dice; the minimum and maximum values are now 2 and 12, which results in 11 different totals (macrostates). Hence, the expression for the number of macrostates should increase as the number of dice increases, and at the same time, consider that the minimum value is not 1 when we have more than 1 dice. This expression represents the correct relationship between the number of dice and the number of macrostates. So, the correct answer is: d) \(5N+1\)

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