Prison Combinatorics

The illustration above shows one fundamental problem.  Suppose two prisoners are to share a single cell.  How many different combinations are there?

If there is only a small prison population, there are few possibilities.  If there are only 100 prisoners, to be placed in 50 cells, then there are only 4,950 possible ways of doing it.

With a larger prison population, there are more combinations.  If there were 500 prisoners to share 250 cells, there are 124,750 ways of distributing them.  But an increase in the prison population does not raise the number of combinations nearly as much as raising the size of the groupings.

Suppose that the 500 prisoners were to be placed in ten cell blocks, each one holding 50 prisoners who ate together in the prison cafeteria and took their exercise together in the prison yard.  Each group of 50 prisoners would get to get to know one another quite well, while having little at all to do with the others in the building.  How many combinations would result.  The number is preposterous, hardly believable.  The number of ways you can divide 500 people into groups of 50 is about 231 followed by 67 zeroes, which is roughly the number of atoms in our galaxy.

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