How can there be more ways to arrange a deck of cards than there are atoms on earth?

Question

I understand the math behind it, I just can’t wrap my head around the fact that something so common and limited like a deck of cards can have more ways to be arranged than something so massive like the earth with all its oceans and mountains has atoms.

In my mind it would make more sense that even a little pond has more atoms than there are deck arrangements.

Could it be due to the fact that atoms have a lot of empty space in them?

In: 318

Answer 1

You are counting two completely different things, both of which are hard to grasp intuitively.

First of all, I think that if I told you there are 10^(100) atoms in Earth you would probably believe me. This number is incredibly off. These numbers are so absurdly massive it is difficult to comprehend, but it is the same overestimation as saying “one Earth” when in reality you have one little atom.

Second, the number of permutations is also counterintuitively massive. Have you heard the story of the checkerboard and the grain of rice? A pharaoh was very please with a wise man, and asked how could he pay. The wise man said he wanted the pharaoh to put one grain of rice in the first square of a checkerboard, two in the second one, four in the third, eight in the fourth… Always doubling it. Each four squares the total becomes 16 times larger. By the end of the second round, you are about 1 kg. By the time you hit half the board, you’ve passed one ton. When you fill the board, you hit 100 times the global production of rice last year. [NOTE: my numbers may be off, but the huge growth stands].

The number of shuffles is 52 for 1 card, 52 × 51 = 2652 for 2 cards, 52 × 51 × 50 = 132 600 for 3 cards… For 52 cards, 52 × 51 × … × 2 × 1 is about 10^(68). It is not exactly the same as with the rice, but similar enough to understand it.

It is hard to wrap you head around this, I know. Adding one element makes things explode. For counting atoms in Earth, one atom is just one atom. Twice as many checkerboard squares will (may) make the amount of rice stupidly larger. Twice as many atoms is “just” twice as many atoms.

This is called *exponential growth*. People say a lot “exponentially large”, but they most often use it wrong. It just does not mean “very big”, it means “this grows much, much, MUCH faster the more something else grows”.