The nation of Griddonesia consists of eighty-one equally-spaced islands represented by intersections of the lines in the following grid, where north is up and east is right as on a standard map. Each island is connected to all its adjacent islands by horizontal and vertical bridges exactly one-mile long. There are no diagonal bridges. A Griddonesian oceanic engineer labels each island with a different integer from 1, 2, ..., 81 to identify them. In how many ways can he number the islands, where the number 1 is always assigned to the central island?
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First, let's look at a simpler problem
How many ways do we have assigning 3 letters (A,B,C) to three boxes?
we have three ways to assign the letter "A":
A _ _ , _ A _ , _ _ A
Once this letter is assigned, we have two empty boxes. Thus we have for each original configuration two ways to assign the ...
The solution explains the way to find teh number of possible permutations out of a sample.