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

一副牌的排序有多少種？ - 楊奈·凱金 (Yannay Khaikin)

譯者: Helen Chang 審譯者: Regina Chu

Pick a card, any card.

抽一張牌，隨便一張，

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

其實，乾脆把整副牌都攤開來 看一看，

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

一副共有 52 張的撲克牌， 已沿用了好幾個世紀。

Everyday, thousands just like it

每天，成千上萬副這樣的牌，

are shuffled in casinos all over the world,

在全球各個賭場被洗來洗去，

the order rearranged each time.

每次洗都會重新排序。

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

但當你每回拿起一副洗好的牌，

like this one,

像這副一樣，

you are almost certainly holding

你幾乎可以確定的是，

an arrangement of cards

你手上這副牌的順序

that has never before existed in all of history.

在過去從未出現。

How can this be?

怎麼會這樣？

The answer lies in how many different arrangements

答案在於，究竟有多少排列組合， 不論是這 52 張牌，

of 52 cards, or any objects, are possible.

或任何物件， 有多少可能的排列組合存在？

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

52 看起來不是個很大的數字，

but let's start with an even smaller one.

但我們還是先從 更小的數字開始吧。

Say we have four people trying to sit

例如有四個人嘗試坐在

in four numbered chairs.

四張有編號的椅子上，

How many ways can they be seated?

他們的座位有幾種坐法？

To start off, any of the four people can sit

一開始，四人中的任何一位 都可以坐在一號位置，

in the first chair.

One this choice is made,

決定好之後，

only three people remain standing.

還有三個人站著，

After the second person sits down,

第二個人坐下之後，

only two people are left as candidates

就剩下兩個人有可能

for the third chair.

坐在三號位置。

And after the third person has sat down,

第三個人坐下後，

the last person standing has no choice

最後一個站著的人便別無他選，

but to sit in the fourth chair.

只能坐在四號椅子。

If we manually write out all the possible arrangements,

如果我們寫下 所有可能的座位排法，

or permutations,

或者說排列， (permutations)

it turns out that there are 24 ways

結果將有 24 種不同的坐法，

that four people can be seated into four chairs,

讓四個人坐上四張椅子。

but when dealing with larger numbers,

但當要處理的數字較大時，

this can take quite a while.

這就要花上好些時間了。

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

我們來想想有沒有更快的方法。

Going from the beginning again,

從頭來過，

you can see that each of the four initial choices

由誰坐上一號椅子，

for the first chair

leads to three more possible choices for the second chair,

引出二號椅子的三種可能選擇，

and each of those choices

而當中的每個選項，

leads to two more for the third chair.

再引出三號座位的兩種可能性。

So instead of counting each final scenario individually,

所我們不需要 一個一個排出最終的坐法，

we can multiply the number of choices for each chair:

只需乘上每張椅子的可能選項： 4 乘以 3 乘以 2 乘以 1。

four times three times two times one

to achieve the same result of 24.

就會得到相同的結果， 即 24 種坐法。

An interesting pattern emerges.

所以，出現了有趣的規則：

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

我們先確認要排列的物件數量，

four in this case,

這次是四個人，

and multiply it by consecutively smaller integers

然後連續乘以越來越小的整數，

until we reach one.

直到 1 為止。

This is an exciting discovery.

這是很有趣的發現， 數學家將這種計算方法

So exciting that mathematicians have chosen

to symbolize this kind of calculation,

known as a factorial,

命名為階乘，

with an exclamation mark.

以驚嘆號「！」表示。

As a general rule, the factorial of any positive integer

一般而言，任意整數的階乘，

is calculated as the product

計算方法為：

of that same integer

從自己開始，越來越小的整數， 往下相乘，直到 1 為止。

and all smaller integers down to one.

In our simple example,

我們剛剛那個簡單的例子，

the number of ways four people

4 個人座位的排列方法，

can be arranged into chairs

is written as four factorial,

就可以寫成 4 的階乘「 4! 」，

which equals 24.

計算結果等於 24。

So let's go back to our deck.

所以讓我們回頭來看這副牌，

Just as there were four factorial ways

如同計算 4 個人 座位的排列方式，

of arranging four people,

there are 52 factorial ways

52 張牌就有 52! 種排列方式。

of arranging 52 cards.

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

好在我們不需要用手算，

Just enter the function into a calculator,

只要按計算機就可以知道，

and it will show you that the number of

可能的排列方式共有

possible arrangements is

8.07 x 10^67,

8.07 乘以 10 的 67 次方 這麼多種的可能排序，

or roughly eight followed by 67 zeros.

大約就是 8 後面加上 67 個 0 。

Just how big is this number?

這數字到底是多大呢？

Well, if a new permutation of 52 cards

嗯，如果每秒鐘排一種順序，

were written out every second

starting 13.8 billion years ago,

大約要花 138 億年，

差不多是從 宇宙大爆炸要開始的時候，

when the Big Bang is thought to have occurred,

the writing would still be continuing today

一直排到此時此刻都還沒排完，

還要再排個幾百萬年， 才可能排出所有的可能順序。

and for millions of years to come.

In fact, there are more possible

事實上，52 張牌的排法，

ways to arrange this simple deck of cards

數量可能遠超過， 地球上所有的原子數目總和。

than there are atoms on Earth.

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

所以下次輪到你洗牌的時候，

take a moment to remember

記得想想 你現在洗出來的這副牌，

that you're holding something that

may have never before existed

它的排列順序， 可能是絕無僅有，空前絕後的。

and may never exist again.