How huge?


OK, let's imagine how many ways we could distribute 64 bean bags (units of energy) into 64 pots (degrees of freedom) such that exactly one bean bag ends up in each pot.

Suppose we throw the first bag, and it drops into a pot. Which one? We don't care. It could be any of the 64, so there are then 64 possible ways we could have one packet in any one pot.

Now we throw the second. It's got to end up in one of the 63 pots not already occupied by our first bag, and there are of course 63 ways in which this could happen. So altogether now there are 64 x 63 = 4,032 possible ways in which we could have the two bean bags each land in different pots.

Well, you can see where we're going. There are 64 x 63 x 62 = 249,984 ways in which we could throw three bean bags and have them land in three different pots. There are 64 x 63 x 62 x 61 = 15,249,024 ways in which we could throw four bean bags and have them land in four different pots, and by the time we're done the number of ways we could throw 64 bean bags and have them land in 64 different pots is 64 x 63 x 62 x 61 x 60 x 59 x 58 x . . . x 4 x 3 x 2 x 1. This horrible number, sometimes abbreviated 64! (read "sixty-four factorial") is equal to:

126,886,932,185,884,164,103,433,389,335,161,480,802,865,516,174,545,192,198,801,894,375,214,704,230,400,000,000,000,000 
or about 1.3 x 1089. This is far more than the number of atoms in the entire Universe.

Furthermore, if you contrast this number with the number of ways in which all 64 bean bags could end up in one particular pot (which is exactly 1), then you can see that the odds of accidentally getting all the energy to end up in one degree of freedom is about 1 in 1089.

Not even the lottery gives you odds that bad.


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