What is 52 Factorial?

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Are you ready to have your mind blown? Here’s what you need to do.

Grab a deck of cards: plain, fancy, laminated, varnished, old or new. Four suits of 13 cards: clubs, hearts, spades, and diamonds. Shuffle them. Riffle them. Cut them. Cut them, again. Now, place them on the table.

That’s it…

In front of you is something utterly unique: a combination of cards, so rare, it will never be repeated.

In fact: if you’d started shuffling that deck at the very start of time, dealing a fresh set of cards every second, you would never hit a matching deck.

Let’s go forward in time. Our sun is collapsing from a red giant star. It’s five billion years into the future. The Earth is long gone. You sit alone in your floating spacecraft. You’re still shuffling that deck; a fresh deal, every second.

Even now: you won’t be remotely close to dealing the same order of cards twice.

52 Factorial: The Numbers

Let’s get mathy.

To get our heads round the colossal majesty of a well-shuffled deck of cards, we need to understand some basic math; in this case, the world of the factorial.

In layman’s terms, a factorial is the number of potential combinations, contained within a particular set of integers.

Let’s use playing cards as an example.

If you hold a single King, K♦, in your hand, it has a factorial of one. It’s written 1!. There is only one possible combination of cards.

Now, you add a Queen. You have two potential combinations, they are:

• K♦ Q♣
• Q♣ K♦

That’s it: that’s 2 factorial or 2!.

Now, let’s make things a little more interesting, by throwing in the Jack. Three cards: here are the possible combinations:

• K♦ Q♣ J
• K♦ J Q♣
• Q♣ K♦ J
• Q♣ J K♦
• JQ♣ K♦
• J K♦ Q♣

So, 3 factorial (3!) equals a total of 6.

Finally, so you can get a true idea how the numbers escalate, we’ll have a look at 4 factorial (4!)

Let’s add a 10 to the deck. Here are the possible combinations:

• K♦ Q♣ J 10♠
• K♦ Q♣ 10♠ J
• K♦ J Q♣ 10♠
• K♦ J 10♠ Q♣
• K♦ 10♠ Q♣ J
• K♦ 10♠J Q♣
• Q♣ K♦ J 10♠
• Q♣ K♦ 10♠ J
• Q♣ J 10♠ K♦
• Q♣ J K♦ 10♠
• Q♣ 10♠ K♦ J
• Q♣ 10♠ J K♦
• J K♦ Q♣ 10♠
• J K♦ 10♠ Q♣
• J Q♣ K♦ 10♠
• J Q♣ 10♠ K♦
• J 10♠K♦ Q♣
• J 10♠Q♣ K♦
• 10♠ K♦ Q♣ J
• 10♠ K♦ J Q♣
• 10♠Q♣ K♦ J
• 10♠ Q♣ J K♦
• 10♠JK♦ Q♣
• 10♠ J Q♣ K♦

So, with just four cards, we have 24 potential combinations. 4 Factorial (4!) equals 24.

If we add just one more card, there are 120 permutations. With six cards, there are 720. Deal seven cards and you have 5,040 variations.

Here’s the thing: with just 10 cards, that’s one hand of gin rummy, you have – *pauses for effect* – a staggering 3,628,800 possible permutations. That’s 3.6 million.

That’s how many different combinations there are of just 10 cards.

It’s a pretty good party trick. Just deal out 10 cards and ask you guests to guess how many different ways they think the 10 cards can be arranged. Unless they know, they won’t guess. It’s worth a wow or two.

How Big is 52 Factorial?

Basically: each additional card is a multiplier. 52 Factorial is 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 – all the way to 52. It’s not a number you’re going to be able to see on a calculator.

52 factorial written out, the actual number, has 68 digits and looks like this:

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

To a single figure of accuracy, it is a 1 followed by 68 zeros:

100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

For context, When calculating the best RTP slots, a theoretical number of spins of 100 million is used.

That is 100,000,000.

Barely a drop in the ocean compared to the number of possible deals a 52-card deck can produce.

But how can anyone possibly conceptualize a number this big? There’s big and then there’s this.

Here are some scenarios that might just give you a little perspective on the scale of 52!.

How Long is 52 Factorial in Seconds?

To get the ball rolling, we start a countdown. We input 52 factorial – the number above – as seconds.

We hit start. We are off and running. The seconds are counting down.

Let’s start on the equator. We begin our journey in Pontianak: the capital of West Kalimantan (West Borneo); the only city in the world that sits exactly on the equator. Grab some lunch and get ready. It’s going to be a long day.

We are now going to walk along the equator. We will circumnavigate the world – one step at a time; a journey of 40,075,017 meters.

But, we’re not in a hurry. In fact: we will take one step, every one billion years.

Let’s hope you’ve got a few good books to read. This is going to take time: 1,000,000,000 (years) x 40,075,017 (meters); a total of 40,075,017,000,000,000 years for one circumnavigation.

Eventually, we return  to Pontianak. It took a while, but we finally got round the planet.

Now, we take a single drop of water from the Pacific Ocean.

And, off we go again: we circumnavigate the globe, one step every billion years. Let’s hope you’ve got a few Netflix box sets lined up.

When we get to Pontianak, for the second time, we take another drop of water.

The Pacific Ocean contains 707.6 million cubic kilometers of water. One milliliter contains 20 drops. It’s going to take a while to empty. But we persist.

Eventually, the Pacific Ocean is empty. It’s time for the next phase.

Get Your Head Around 52! (Or Try To)

Now, we lay a single sheet of paper on the ground.

We refill the Pacific Ocean.

We go back to square one.

The timer is still counting down.

Off we go again: one step – one meter – every one billion years. Circumnavigate the planet. Remove one drop of water. Empty the Pacific Ocean, for a second time.

We lay a second sheet of paper on top of the first.

Still with it?

What we have to do is stack those sheets of paper up – really high. In fact: we need to pile the paper up, until it reaches the Sun: a distance of 149,597,870.691 kilometers.

That’s one sheet of paper – every time the Pacific Ocean is emptied.

Good news: finally, we reach the sun.

The timer is still counting down.

Here’s the bad news: even if you repeat this process 1,000 times, you are still not even a third of the way there.

The cards are shuffling; one fresh deal every second and you are still nowhere near exhausting all the possible combinations.

Bored of circumnavigating the earth?

Let’s try something different. After all: we still have lots of time left to kill. 52 Factorial has barely kicked in.

Let’s shuffle those cards.

52 Factorial and Poker

Deal yourself a hand – once every one billion years. Be patient. Have you dealt yourself a Royal Flush? No? Not a problem: wait one billion years and deal yourself another hand.

Is it a Royal Flush? A Royal Flush hits about once in every 649,740 hands hence it’s place atop poker hand rankings.

Keep dealing yourself a five-card hand, once every billion years. When you hit a Royal Flush, it’s time to buy a lottery ticket. Have you won the jackpot? No? Unlucky: start again. The odds of hitting a six number (1- 59) lottery win are 45,057,474 to 1.

Finally, you hit a Royal Flush. You buy a lottery ticket. You win the jackpot.

Now, you have earned the right to throw a single grain of sand into the Grand Canyon.

Repeat this process: win the Royal Flush, win the lottery, throw in a grain of sand.

When the Grand Canyon is completely full, you remove one ounce of rock from Mt Everest.

The Grand Canyon empties and you start again. Repeat the entire process.

The timer is still counting down. Still, you are nowhere near the number of potential combinations of a 52-card deck.

When you have finally levelled Mt Everest, ounce by ounce, Grand Canyon by Grand Canyon, you still have more time to kill.

The truth is: you need to level Mt Everest 256 times to run the clock to zero. All this after 1,000 trips to the Sun.

That’s how many different combinations there are in a single deck of cards.

And you thought betting a 100/1 outsider with an online sportsbook was taking a big chance.

52! and the Big Bang

Of course, none of this will ever happen. Just five steps into your first circumnavigation of the globe, the Sun will run out of hydrogen.

You’ll never get to remove that first drop of water. Let alone, empty the ocean, build your paper tower to the stars, fill (and empty) the Grand Canyon with sand and level Mt Everest 256 times.

Here’s the thing. No one knows precisely how long the universe has left. Our cosmos is currently a sprightly 13.77 billion years old. Stars are still being born. It’s estimated that the last star will be born one trillion years from now. It will probably be a small Red Dwarf called Dave.

That Red Dwarf will shine, fade, and then twinkle in the gloom. In roughly 100 trillion years, the lights will finally go out.

So, we have one hundred trillion years to get through this deck of cards.

One hundred trillion looks like this, a one with 14 zeros: 100 000 000 000 000

Multiply this by days of the year (365): 36 500 000 000 000 000

Multiply this by hours in a day (24): 876 000 000 000 000 000

Again, by minutes (60): 52 560 000 000 000 000 000

Finally, by seconds (60): 3 153 600 000 000 000 000 000

Now, take another look at that first factorial 52!

Let’s divide 52! By the number of seconds remaining in the universe; assuming one new deal per second

The stars would have to be born and die, this many times, before you had revealed every combination in that single, simple, 52-card deck:

25 576 539 564 606 760 074 727 497 734 780 494 347 821 380 467

Mind blown? It’s all in the cards… So deal ‘em out at an online poker site and enjoy something special…

References

Paul Cullen
Casino Industry Expert
Paul Cullen
Casino Industry Expert

Paul Cullen is an industry veteran, with a track record that stretches back to day one. He started his career as a copywriter and creative for the world’s very first online sportsbook: Intertops.com. There was no one else. Since then, he has seen the industry evolve and grow, working at BetonSports, BetWWTS, Absolute Poker, Ultimate Bet, InterCasino, PartyGaming, Mansion, Bodog, Casino Choice, Costa Bingo and Casumo. The evolution of Internet gaming, the arrival of the online casino, the poker revolution, and the bingo boom. He’s got the t-shirt.