Question

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

Based on the given step-by-step solution, the correct answer for finding the number of macrostates resulting from rolling a set of N six-sided dice is a) \(6^{N}\). This formula takes into account all possible combinations of dice rolls, as for each die, there are 6 choices or possibilities independently.

Step-by-step solution

Understand the problem

The first thing we need to do in this problem is to understand the concept of macrostates in this context. A macrostate is the sum of the pips on top of the N dice, so the answer must take into account all possible combinations of dice rolls.

Analyze the given options

We have to figure out which of the given formulae would determine the number of macrostates. a) \(6^N\): This formula calculates the total number of outcomes when rolling N dice, each with 6 sides. It multiplies 6 by itself N times. b) \(6N\): This formula would find us the total number of pips that can be found on N dice, it does not calculate the number of macrostates. c) \(6N-1\): This formula is the subtraction of 1 pip from all possible pips in case N. It seems not relevant to the context of the problem. d) \(5N+1\): This calculation adds 1 pip for every five pips, so it’s actually incrementing the total pips but does not find the number of macrostates possible.

Choose the correct formula

The only option that takes into account all possible combinations of rolls is a) \(6^N\). This is because for each of the N dice, we have 6 choices or possibilities, independently. So, for N dice, there would be \(6^N\) possible outcomes (equivalent to the number of ways we could add the pips on the N faces to find macrostates).

Answer

The correct answer for the number of macrostates resulting from rolling a set of \(N\) six-sided dice is: a) \(6^{N}\).