Professor Libbrecht from Caltech was kind enough to explain the snowflake problem to me, and to give me permission to quote him.
Let’s make the numbers a bit smaller to make things clearer. Say a snowflake contains just 10,000 molecules, and these lined up in a straight line. Most of these are ordinary water molecules, but let’s say 100 are heavier isotopes.
Then the question is, how many ways can you arrange these 100 heavier molecules in this linear crystal? Well, the first heavy molecule could go at position 1, or 2, or any position up to number 10,000. Thus there are 10,000 places to substitute the first heavy molecule.
Similarly, the second heavy molecule has 9,999 possible locations. If you placed just two heavy molecules in the crystal, there are already 99,990,000 different ways to do it. By the time you place all 100, there are nearly 10^400 different ways to arrange things.
Since 10^400 is a very large number, there is basically no chance you would place 100 molecules in the same way twice.
The same logic applies to snow crystals, except all the numbers are much, much larger.