How unique is a pack of cards?

Every household owns a deck of cards: whether used for games nights, magic tricks or just to build a tower. When you shuffle your pack of cards, just how different is it from when you last picked it up, or the person next to you?

A pack of cards is made up of 52 distinctive cards, sorted by 2 colours, 4 suits and 13 different cards in each suit. That does not seem like a lot of unique cards, but there are plenty of ways to order them… So just how can we find out how many ways there are to shuffle said cards? The beauty of a pack of cards is the endless number of times you can compete; each time you give the cards a shuffle, a new order is created, and a new game can proceed. Knowing this we can apply a maths function called ‘factorial’ to our pack of cards and mathematically work out how many ways you can order the pack of cards, or how many times you can play.

The factorial is a function, applied to any number $n$, denoted $n!$, and can be found on any scientific calculator – have a look for it on yours! It takes the form $n! = n \times (n-1) \times (n-2) \times … \times 1$, or in English, it is the product of all the (positive) integers less than and equal to $n$.

For example, $3! = 3 \times 2 \times 1$, which is equal to $6$. 

But how is this useful to us? We can use the factorial function to work out the number of arrangements for a given number of elements, without any repetitions. As an example, let’s visualise 3 letters: A, B and C. We want to work out how many unique arrangements we have of these. There are 3 choices for the first letter of our sequence, 2 choices for the second, leaving 1 choice for the last letter. So overall, we have $3\times2\times1=3!=6$ different arrangements. Thus, we have $n!$ different arrangements of $n$ elements.

Applying this function to our cards we get $52!$. Try this on your calculator- a very big number, right? The answer is around $8.065817519\times10^{67}$ different arrangements of a pack of cards, in fact. That is a very long number, and to put this into perspective lets give some other statistics:

  • The world’s population is around $7.339 \times 10^9$ people.
  • There is approximately $1.260 \times 10^{21}$ litres of water on the planet earth.

That’s a lot less than the number of combinations from a simple pack of cards.

We can relate these statistics to what we can do with pack of cards. Imagine we have every single person on the planet simultaneously shuffling cards, and we want to create every single possible combination. How many packs of cards will each person need? We use simple division:

$$ \frac{8.065817519 \times 10^{67}}{7.339 \times 10^9}$$

To get our answer of $1.099035 \times 10^{58}$ packs of cards each, or in words ‘ten octodecillion nine hundred ninety septendecillion three hundred fifty sexdecillion’ packs of cards. Try ordering that many packs on Amazon!

Let us take this further. Atoms make up everything, including water and humans, and are the smallest unit of matter known to scientists. They make up objects and organisms by having enormous amounts atoms in a space until they form rigid structure. In just one human there is around $7 \times 10^{27}$ atoms – a lot of matter for just one person! Okay, what about something bigger… like the sun? It is the biggest star in our solar system, and is made up of hydrogen atoms – that’s why it burns! We can see the sun everyday miles and miles away and it is burning to create heat for us to survive. Inside is a huge number of around $1 \times 10^{57}$ hydrogen atoms. This is still just one tenth of the number of packs of cards we said to order above! So now we can compare our little pack of playing cards to the sun. Remarkable!

That just signifies there is still more ways to order a pack of cards than there are atoms making up our whole world as we know it. A crazy statistic and really shows how understated a pack of cards really is.

Subsequently, to answer our question of ‘how unique is a pack of cards?’, we can say VERY. Next time you pick up a pack of cards and go to shuffle them, just consider how many ways you can do it. Think, how likely is it you have the same combination as someone else? A very small chance, so appreciate the uniqueness of the cards you are holding.

Georgia is in her second year of Mathematics at UEA and will be completing her final year from September. She started her university career studying Chemical Physics, but found herself more partial to Maths. She loves the link between maths and science and how it explains the real world (just look at her articles!). When she's not studying she can be found out and about, eating and drinking in Norwich and London.