https://czep.net/weblog/52cards.html

52 Factorial

It Starts with a Simple Deck of Playing Cards

They seem harmless enough, 52 thin slices of laminated cardboard with
colorful designs printed on their sides. Yet, as another illustration
of the mantra that complexity begins from the most simple systems,
the number of variations that these 52 cards can produce is virtually
endless. The richness of most playing card games owes itself to this
fact.

Permute this!

The number of possible permutations of 52 cards is 52!. I think the
exclamation mark was chosen as the symbol for the factorial operator
to highlight the fact that this function produces surprisingly large
numbers in a very short time. If you have an old school pocket
calculator, the kind that maxes out at 99,999,999, an attempt to
calculate the factorial of any number greater than 11 results only in
the none too helpful value of "Error". So if 12! will break a typical
calculator, how large is 52!?

52! is the number of different ways you can arrange a single deck of
cards. You can visualize this by constructing a randomly generated
shuffle of the deck. Start with all the cards in one pile. Randomly
select one of the 52 cards to be in position 1. Next, randomly select
one of the remaining 51 cards for position 2, then one of the
remaining 50 for position 3, and so on. Hence, the total number of
ways you could arrange the cards is 52 * 51 * 50 * ... * 3 * 2 * 1,
or 52!. Here's what that looks like:

80658175170943878571660636856403766975289505440883277824000000000000

This number is beyond astronomically large. I say beyond
astronomically large because most numbers that we already consider to
be astronomically large are mere infinitesmal fractions of this
number. So, just how large is it? Let's try to wrap our puny human
brains around the magnitude of this number with a fun little
theoretical exercise. Start a timer that will count down the number
of seconds from 52! to 0. We're going to see how much fun we can have
before the timer counts down all the way.

Shall we play a game?

Start by picking your favorite spot on the equator. You're going to
walk around the world along the equator, but take a very leisurely
pace of one step every billion years. The equatorial circumference of
the Earth is 40,075,017 meters. Make sure to pack a deck of playing
cards, so you can get in a few trillion hands of solitaire between
steps. After you complete your round the world trip, remove one drop
of water from the Pacific Ocean. Now do the same thing again: walk
around the world at one billion years per step, removing one drop of
water from the Pacific Ocean each time you circle the globe. The
Pacific Ocean contains 707.6 million cubic kilometers of water.
Continue until the ocean is empty. When it is, take one sheet of
paper and place it flat on the ground. Now, fill the ocean back up
and start the entire process all over again, adding a sheet of paper
to the stack each time you ve emptied the ocean.

Do this until the stack of paper reaches from the Earth to the Sun.
Take a glance at the timer, you will see that the three left-most
digits haven t even changed. You still have 8.063e67 more seconds to
go. 1 Astronomical Unit, the distance from the Earth to the Sun, is
defined as 149,597,870.691 kilometers. So, take the stack of papers
down and do it all over again. One thousand times more.
Unfortunately, that still won t do it. There are still more than
5.385e67 seconds remaining. You re just about a third of the way
done.

And you thought Sunday afternoons were boring

To pass the remaining time, start shuffling your deck of cards. Every
billion years deal yourself a 5-card poker hand. Each time you get a
royal flush, buy yourself a lottery ticket. A royal flush occurs in
one out of every 649,740 hands. If that ticket wins the jackpot,
throw a grain of sand into the Grand Canyon. Keep going and when you 
ve filled up the canyon with sand, remove one ounce of rock from Mt.
Everest. Now empty the canyon and start all over again. When you ve
levelled Mt. Everest, look at the timer, you still have 5.364e67
seconds remaining. Mt. Everest weighs about 357 trillion pounds. You
barely made a dent. If you were to repeat this 255 times, you would
still be looking at 3.024e64 seconds. The timer would finally reach
zero sometime during your 256th attempt. Exercise for the reader: at
what point exactly would the timer reach zero?

Back here on the ranch

Of course, in reality none of this could ever happen. Sorry to break
it to you. The truth is, the Pacific Ocean will boil off as the Sun
becomes a red giant before you could even take your fifth step in
your first trek around the world. Somewhat more of an obstacle,
however, is the fact that all the stars in the universe will
eventually burn out leaving space a dark, ever-expanding void
inhabited by a few scattered elementary particles drifting a tiny
fraction of a degree above absolute zero. The exact details are still
a bit fuzzy, but according to some reckonings of The Reckoning, all
this could happen before you would've had a chance to reduce the vast
Pacific by the amount of a few backyard swimming pools.


The Details

Please be advised that rounding and measurement error combined are
many orders of magnitude greater than the current age of the
universe, 4.323e17 seconds.

  * 52! is approximately 8.0658e67. For an exact representation, view
    a factorial table or try a "new-school" calculator, one that
    understands long integers.
  * A billion years currently equals 3.155692608e16 seconds; however,
    the addition of leap seconds due to the deceleration of Earth's
    orbit introduces some variation.
  * The equatorial circumference of the Earth is 40,075,017 meters,
    according to WGS84.
  * One trip around the globe will require a bit more than 1.264e24
    seconds, assuming 1 meter per step, which is actually quite a
    stretch for most people. This is almost 3 million times the
    current age of the universe, and we still have 2 levels of
    recursion to go (ocean, stack of papers).
  * There are 20 drops of water per milliliter, and the Pacific Ocean
    contains 707.6 million cubic kilometers of water, which equals
    about 1.4152e25 drops.
  * 1 Astronomical Unit, the distance from the Earth to the Sun, is
    defined as 149,597,870.691 kilometers.
  * A royal flush occurs in one out of every 649,740 hands.
  * The odds of winning a lotto jackpot after matching 6 numbers
    chosen without replacement from the range 1 to 59 are 1 in
    45,057,474.
  * The Grand Canyon has an estimated volume of 40 billion cubic
    meters. 1 grain of sand occupies approximately 1 cubic milimeter.
    Thus, the Grand Canyon could hold roughly 4e19 grains of sand.
  * This article estimates that Mt. Everest weighs about 357 trillion
    pounds.
  * Here you can read all about The End.