One Deck to Rule Them All

Want to know the power of mathematics?

I was recently reminded of one of my favourite mathematical curiosities and thought I’d share it with you as it never fails to grab attention and get people thinking.

If you properly shuffle a standard deck of 52 playing cards, then the chances are that the arrangement of cards you are now holding has never existed before in the history of the human race and likely wont again.

Read that again…

That’s pretty awe inspiring when you think about it.

The reason I like this fact is its something that you can connect with pretty easily via a deck of playing cards, but it also seems so unlikely at first glance, until you delve into the mathematics.

So, lets do that now, and then you’ll hopefully appreciate why this means that human card dealers almost certainly create a unique deck with every shuffle.

It’s actually fairly simple maths. There are 52 unique cards in a single deck. Each card can appear in any of 52 positions. Once the first card is in place, there are only 51 remaining possibilities, then 50 and so on. So you can think of this as:

52 x 51 x 50 x 49… … x 3 x 2 x 1 different ways. This can be represented as 52!

52! in mathematics is read as ‘fifty two factorial’, and it means the above calculation which gives you every possible arrangement of cards that a deck can take.

So what’s the big deal? Well if you do that calculation, what you get is a number that is so big it has 67 zeros after it (approximately 8 x 10⁶⁷ actually) or if you prefer:

80000000000000000000000000000000000000000000000000000000000000000000

To give that some perspective that is significantly larger than the number of atoms on Earth – that’s a number with about 50 zeros after it.

So even if every human who has ever lived had shuffled a deck every second of their life, we still would be nowhere near all the possible permutations. That is why any particular shuffle is, with overwhelming probability, unique in the history of the universe!

Cool eh?