How can we correct for this overcounting? How do we "remove the
labels" from the atoms? As a correction factor, how many different
label shufflings can be made for each arrangement of four atoms?
Label ‘a’ could be given to any one of the four atoms, label ‘b’
to any of the remaining three, label ‘c’ to two, and label ‘d’ then
has to go to the last atom. There are 4 * 3 * 2 * 1 = 24 meaningless
permutations of labels for every really different arrangement of
atoms. We have overcounted by a multiplicative factor of 24.
Hence the 3024 ways of arranging atoms must be divided by 24 to
remove the labels on the atoms. The number of different ways of
arranging four indistinguishable atoms among nine locations
Of the 126 possible arrangements only four are crystals, and the
other 122 lead to a gas. Even in such a tiny and restricted universe,
a gas is far more likely to result from a random arrangement of
atoms than is a crystal. This is true because the specifications
for a crystal are so much more restrictive:
Crystal - four adjacent atoms in a square;
Gas - four atoms in any arrangement except those that lead to a
Since the atoms do not have names or
labels, and all look alike, all of the above pictures correspond
to the same single atomic arrangement. That is four atoms at positions
3, 4, 7, and 8.