0:00:06.954,0:00:09.124
Pick a card, any card.

0:00:09.124,0:00:12.014
Actually, just pick up all of them and take a look.

0:00:12.014,0:00:15.848
This standard 52 card deck has been used for centuries.

0:00:15.848,0:00:18.098
Everyday, thousands just like it

0:00:18.098,0:00:21.134
are shuffled in casinos all over the world,

0:00:21.134,0:00:23.719
the order rearranged each time.

0:00:23.719,0:00:26.431
And yet, every time you pick up a well shuffled deck

0:00:26.431,0:00:27.642
like this one,

0:00:27.642,0:00:29.431
you are almost certainly holding

0:00:29.431,0:00:30.848
an arrangement of cards

0:00:30.848,0:00:33.729
that has never before existed in all of history.

0:00:33.729,0:00:35.764
How can this be?

0:00:35.764,0:00:37.900
The answer lies in how many different arrangements

0:00:37.900,0:00:42.348
of 52 cards, or any objects, are possible.

0:00:42.348,0:00:45.620
Now, 52 may not seem like such a high number,

0:00:45.620,0:00:48.035
but let's start with an even smaller one.

0:00:48.035,0:00:49.932
Say we have four people trying to sit

0:00:49.932,0:00:52.348
in four numbered chairs.

0:00:52.348,0:00:54.460
How many ways can they be seated?

0:00:54.460,0:00:56.598
To start off, any of the four people can sit

0:00:56.598,0:00:57.920
in the first chair.

0:00:57.920,0:00:59.132
One this choice is made,

0:00:59.132,0:01:01.466
only three people remain standing.

0:01:01.466,0:01:03.262
After the second person sits down,

0:01:03.262,0:01:05.219
only two people are left as candidates

0:01:05.219,0:01:06.680
for the third chair.

0:01:06.680,0:01:08.680
And after the third person has sat down,

0:01:08.680,0:01:10.431
the last person standing has no choice

0:01:10.431,0:01:12.347
but to sit in the fourth chair.

0:01:12.347,0:01:15.098
If we manually write out all the possible arrangements,

0:01:15.098,0:01:16.814
or permutations,

0:01:16.814,0:01:18.818
it turns out that there are 24 ways

0:01:18.818,0:01:22.180
that four people can be seated in to four chairs,

0:01:22.180,0:01:23.991
but when dealing with larger numbers,

0:01:23.991,0:01:25.532
this can take quite a while.

0:01:25.532,0:01:27.848
So let's see if there's a quicker way.

0:01:27.848,0:01:29.286
Going from the beginning again,

0:01:29.286,0:01:31.370
you can see that each of the four initial choices

0:01:31.370,0:01:32.682
for the first chair

0:01:32.682,0:01:35.999
leads to three more possible choices for the second chair,

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and each of those choices

0:01:37.461,0:01:39.847
leads to two more for the third chair.

0:01:39.847,0:01:43.181
So instead of counting each final scenario individually,

0:01:43.181,0:01:46.262
we can multiply the number of choices for each chair:

0:01:46.262,0:01:49.096
four times three times two times one

0:01:49.096,0:01:51.848
to achieve the same result of 24.

0:01:51.848,0:01:53.681
An interesting pattern emerges.

0:01:53.681,0:01:56.729
We start with the number of objects we're arranging,

0:01:56.729,0:01:58.098
four in this case,

0:01:58.098,0:02:00.847
and multiply it by consecutively smaller integers

0:02:00.847,0:02:02.902
until we reach one.

0:02:02.902,0:02:04.514
This is an exciting discovery.

0:02:04.514,0:02:06.449
So exciting that mathematicians have chosen

0:02:06.449,0:02:08.575
to symbolize this kind of calculation,

0:02:08.575,0:02:10.345
known as a factorial,

0:02:10.345,0:02:12.038
with an exclamation mark.

0:02:12.038,0:02:15.514
As a general rule, the factorial of any positive integer

0:02:15.514,0:02:17.416
is calculated as the product

0:02:17.416,0:02:18.876
of that same integer

0:02:18.876,0:02:21.836
and all smaller integers down to one.

0:02:21.836,0:02:23.263
In our simple example,

0:02:23.263,0:02:24.596
the number of ways four people

0:02:24.596,0:02:26.181
can be arranged in to chairs

0:02:26.181,0:02:28.052
is written as four factorial,

0:02:28.052,0:02:29.975
which equals 24.

0:02:29.975,0:02:31.808
So let's go back to our deck.

0:02:31.808,0:02:33.598
Just as there were four factorial ways

0:02:33.598,0:02:35.431
of arranging four people,

0:02:35.431,0:02:37.598
there are 52 factorial ways

0:02:37.598,0:02:40.014
of arranging 52 cards.

0:02:40.014,0:02:43.066
Fortunately, we don't have to calculate this by hand.

0:02:43.066,0:02:45.014
Just enter the function in to a calculator

0:02:45.014,0:02:46.431
and it will show you that the number of

0:02:46.431,0:02:47.931
possible arrangements is

0:02:47.931,0:02:52.368
8.07 x 10^67,

0:02:52.368,0:02:55.788
or roughly eight followed by 67 zeros.

0:02:55.788,0:02:57.458
Just how big is this number?

0:02:57.458,0:02:59.708
Well, if a new permutation of 52 cards

0:02:59.708,0:03:01.752
were written out every second

0:03:01.752,0:03:04.378
starting 13.8 billion years ago,

0:03:04.378,0:03:06.344
when the big bang is thought to have occurred,

0:03:06.344,0:03:09.094
the writing would still be continuing today

0:03:09.094,0:03:11.676
and for millions of years to come.

0:03:11.676,0:03:13.426
In fact, there are more possible

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ways to arrange this simple deck of cards

0:03:16.345,0:03:18.593
than there are atoms on Earth.

0:03:18.593,0:03:20.759
So the next time it's your turn to shuffle,

0:03:20.759,0:03:22.093
take a moment to remember

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that you're holding something that

0:03:23.174,0:03:25.235
may have never before existed

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and may never exist again.