How many ways can four points be arranged in the plane so that the
six distances between pairs of points take on only two different values?
Many crypto doorknob locks use doors with five buttons numbered
from 1 to 5. Legal combinations allow the buttons to be pushed in specific
order either singly or in pairs without pushing any button more than once.
Thus [(12), (34)] = [(21), (34)]; [(1), (3)]; and [(2), (13), (4)] are legal combinations
while [(1), (14)]; [(134)]; and [(13), (14)] are not.
(a) How many legal combinations are there?
(b) If a sixth button were added, how many legal combinations would
there be?
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