How many ways can you arrange a deck of cards? - Yannay Khaikin

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One deck. Fifty-two cards. How many arrangements? Let's put it this way: Any time you pick up a well shuffled deck, you are almost certainly holding an arrangement of cards that has never before existed and might not exist again. Yannay Khaikin explains how factorials allow us to pinpoint the exact (very large) number of permutations in a standard deck of cards.

Lesson by Yannay Khaikin, animation by The Moving Company Animation Studio.

Pick a card, any card.

Actually, just pick up all of them and take a look.

This standard 52 card deck has been used for centuries.

Everyday, thousands just like it

are shuffled in casinos all over the world,

the order rearranged each time.

And yet, every time you pick up a well shuffled deck

like this one,

you are almost certainly holding

and arrangement of cards

that has never before existed in all of history.

How can this be?

The answer lies in how many different arrangements

of 52 cards, or any objects, are possible.

Now, 52 may not seem like such a high number,

but let's start with an even smaller one.

Say we have four people trying to sit

in four numbered chairs.

How many ways can they be seated?

To start off, any of the four people can sit

in the first chair.

One this choice is made,

only three people remain standing.

After the second person sits down,

only two people are left as candidates

for the third chair.

And after the third person has sat down,

the last person standing has no choice

but to sit in the fourth chair.

If we manually write out all the possible arrangements,

or permutations,

it turns out that there are 24 ways

that four people can be seated in to four chairs,

but when dealing with larger numbers

this can take quite a while.

So let's see if there's a quicker way.

Going from the beginning again,

you can see that each of the four initial choices

for the first chair

leads to three more possible choices for the second chair,

and each of those choices

leads to two more for the third chair,

so instead of counting each final scenario individually,

we can multiply the number of choices for each chair:

four times three times two times one

to achieve the same result of 24.

An interesting pattern emerges.

We start with the number of objects we're arranging,

four in this case,

and multiply it by consecutively smaller integers

until we reach one.

This is an exciting discovery.

So exciting that mathematicians have chosen

to symbolize this type of calculation,

known as a factorial,

with an exclamation mark.

As a general rule, the factorial of any positive integer

is calculated as the product

of that same integer

and all smaller integers down to one.

In our simple example,

the number of ways four people

can be arranged in to chairs

is written as four factorial,

which equals 24.

So let's go back to our deck.

Just as there were four factorial ways

of arranging four people,

there are 52 factorial ways

of arranging 52 cards.

Fortunately, we don't have to calculate this by hand.

Just enter the function in to a calculator

and it will show you that the number of

possible arrangements is

8.07 x 10^67,

or roughly eight followed by 67 zeros.

Just how big is this number?