Really, Really, Really Big Numbers

I love a mind boggling math fact, and Cafe Hayek's Don Boudreaux has a very nice one.

The economist Paul Romer notes the astonishing fact that if you thoroughly shuffle a deck of 52 cards, chances are practically 100 percent that the resulting arrangement of cards has never before existed.

Never.

Each time you shuffle a deck, you produce an arrangement of cards that exists for the first and only time in history. The arithmetic works that way. For a very small number of items, the number of possible arrangements -- which item is first, which item is second, which is third, and so on -- is small. Three items, for example, can be arranged only six different ways. But the number of possible arrangements grows very large very quickly. The number of ways to arrange five items is 120. For 10 items it's 3,628,800. For 15 items it's 1,307,674,368,000. The number of different ways to arrange 52 items is 8.066 times 10 to the 67th power.

This number is so enormous that no human can comprehend it. By way of comparison, the number of ways to arrange a mere 20 items is 2,432,902,008,176,640,000 -- a number larger than the number of seconds that elapse in the course of 10 billion years. And this number is microscopic compared to 8.066 times 10 to the 67th power.

He then goes on to note that if 52 items can be arranged in that many possible ways, the number of ways the resources of any economy can be organized is orders of magnitude higher. From this he concludes that it is insane to hope an economy can be ordered optimally by central planners.

This puts me in the awkward position of agreeing with his conclusion completely, but thinking his argument is not very convincing. A dedicated central planner will respond with any number of variations on "But these are smart people," and "This just shows that we can't leave this organizing to random market forces." Some will even argue that, yes, if you want efficiency the market is the way to go, but we care about more important issues than mere material production, we care about justice and equality.

That in no way diminishes how cool these numbers are, or how stupid central planners really are.