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Ludwig Boltzmann killed himself so that we could do thermodynamics by enumerating the states of a system. Learn to count, the most important skill in statistical mechanics.

There are counters, which are blue on one side and red on the other. If the macroscopic description of our state of interest is \(8\) red on the \(4\times 4\) checkerboard, how many corresponding micro-states are there?

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There are \( 15 \) pigeon-holes for \( 2 \) pigeons. If at most one pigeon can fit inside each hole, how many distinct arrangements are there for the \( 2 \) pigeons to fit inside the \( 15\) holes?

Assume that we can distinguish each pigeon from the others.

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There are six distinct boxes, each of which contains a single ball. If there are three red balls and three blue balls, how many distinct arrangements are possible?

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There are \( 5 \) pigeon-holes for \( 5 \) pigeons. Three of the pigeons are white, and the other two are grey. If each hole can hold at most a single pigeon, how many micro-states are there for the \( 5 \) pigeons to fit into the 5 hole?

Assume that we can distinguish each pigeon by their color, but cannot distinguish two pigeons of the same color.

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