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

事實上，只需將所有牌拿起來看看，

This standard 52-card deck has been used for centuries.

這標準的 52 張卡片已經使用了好幾個世紀。

Everyday, thousands just like it

每天都有成千上萬的人們，

are shuffled in casinos all over the world,

在世界各地的賭場中洗牌。

the order rearranged each time.

每次所有牌都會重新排序。

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

每次你拿起剛洗好的牌，

like this one,

像這個，

you are almost certainly holding

你肯定會拿著，

an arrangement of cards

在歷史上

that has never before existed in all of history.

從沒重複過的一組牌。

How can this be?

怎麼會這樣呢？

The answer lies in how many different arrangements

這個答案在於 52 張牌，

of 52 cards, or any objects, are possible.

或是其他東西的排列方式可能有多少種。

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

現在，52 看起來似乎不是很大的數字，

but let's start with an even smaller one.

那我們從較小的數字開始說起。

Say we have four people trying to sit

有四個人，

in four numbered chairs.

試著要坐在四張椅子。

How many ways can they be seated?

他們有幾種坐下的方式？

To start off, any of the four people can sit

一開始，四個人中的任何人，

in the first chair.

都可以坐在第一張椅子上。

Once this choice is made,

一旦決定好第一個位置後，

only three people remain standing.

只有三個人還站著。

After the second person sits down,

在第二個人坐下後，

only two people are left as candidates

只有兩個人可以選擇

for the third chair.

第三張椅子，

And after the third person has sat down,

當第三個人坐下後，

the last person standing has no choice

最後一個站著的人沒有選擇權，

but to sit in the fourth chair.

只能坐在第四張椅子。

If we manually write out all the possible arrangements,

如果我們手寫列出所有可能的組合，

or permutations,

或排列，

it turns out that there are 24 ways

四個人坐四張椅子的組合，

that four people can be seated into four chairs,

算出來有 24 種，

but when dealing with larger numbers,

但當處理較大數字時，

this can take quite a while.

可能需要花多一點時間計算。

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

所以我們看看是否有更快的方法。

Going from the beginning again,

再從一開始看起，

you can see that each of the four initial choices

你會發現第一張椅子，

for the first chair

有四個人可以選擇坐上去，

leads to three more possible choices for the second chair,

導致第二張椅子只有三個人可以選擇坐上去，

and each of those choices

前兩個選擇完，

leads to two more for the third chair.

第三張椅子只有兩個人可以選擇坐上去。

So instead of counting each final scenario individually,

我們可以將每張椅子的可被選擇的機會相乘，

we can multiply the number of choices for each chair:

而不是單獨計算，

four times three times two times one

4 x 3 x 2 x 1，

to achieve the same result of 24.

得出的結果一樣是 24。

An interesting pattern emerges.

一個有趣的模式出現了。

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

我們從排列的對象數開始排列，

four in this case,

在這裡有四個，

and multiply it by consecutively smaller integers

並且乘以連續且較小的整數，

until we reach one.

直到我們乘到 1。

This is an exciting discovery.

這是個有趣的發現。

So exciting that mathematicians have chosen

數學家

to symbolize this kind of calculation,

將這種運算方式，

known as a factorial,

稱為階乘

with an exclamation mark.

用驚嘆號表示。

As a general rule, the factorial of any positive integer

基本的規則就是，

is calculated as the product

將任何正整數的階乘，

of that same integer

計算為該整數，

and all smaller integers down to one.

與所有較小的整數，直到 1 的乘積。

In our simple example,

在這個簡單的例子中，

the number of ways four people

四個人可以坐在椅子

can be arranged into chairs

的排列數量，

is written as four factorial,

寫成四個階乘，

which equals 24.

也就等於 24。

So let's go back to our deck.

現在回到撲克牌這裡，

Just as there were four factorial ways

就如同四個人的排列，

of arranging four people,

四階乘運算，

there are 52 factorial ways

排列 52 張牌，

of arranging 52 cards.

有 52! 種方式。

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

幸運地，我們可以不用手算，

Just enter the function into a calculator,

只要將公式運算到計算機，

and it will show you that the number of

它將會顯示

possible arrangements is

可能的排列組合，

8.07 x 10^67,

是 8.07 x 10 的 67 次方。

or roughly eight followed by 67 zeros.

或是約 8 的後面在加 67 個 0。

Just how big is this number?

這數字到底有多大？

Well, if a new permutation of 52 cards

如果從 138 億年前開始，

were written out every second

每秒可以

starting 13.8 billion years ago,

寫出一種新的排列方法，

when the Big Bang is thought to have occurred,

當人們認為發生大爆炸時，

the writing would still be continuing today

仍可以持續寫至今天，

and for millions of years to come.

直到數百萬年後。

In fact, there are more possible

事實上，比起地球上的原子，

ways to arrange this simple deck of cards

還有更多撲克牌

than there are atoms on Earth.

的可能排列組合。

So the next time it's your turn to shuffle,

所以下一次換你洗牌時，

take a moment to remember

花一點時間去想，

that you're holding something that

你拿到的牌，

may have never before existed

可能是從未發生過的組合，

and may never exist again.

可能也不會再出現了。