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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.

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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