Classical Mechanics

# Microstates and macrostates

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?

A coin is tossed $$8$$ times, and it is noted that the number of heads is $$3.$$ How many sequences of coin flips could have led to this result?

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.

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?

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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