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

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

翻译人员: Biyue碧玥 Wang王 校对人员: Lanfu Zhang

Pick a card, any card.

选一张牌，任何牌。

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

事实上，把它们全部拿起来看一看

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

标准的 52 张牌已经延用了几个世纪之久。

Everyday, thousands just like it

每天成千上万像这样的扑克牌

are shuffled in casinos all over the world,

在世界各地的赌场中洗牌

the order rearranged each time.

每一次排列组合都会改变

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

事实上， 每一次你从洗过的牌堆里抽一张牌

like this one,

像这样，

you are almost certainly holding

几乎可以肯定你拥有的牌

an arrangement of cards

的排列组合顺序

that has never before existed in all of history.

在历史上从未出现过

How can this be?

为什么是这样？

The answer lies in how many different arrangements

答案藏在这52张牌有

of 52 cards, or any objects, are possible.

许多可能的排列组合

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

现在，52 看起并不是一个大数字

but let's start with an even smaller one.

让我们从一个更小的数字开始研究。

Say we have four people trying to sit

假设有四个人要坐

in four numbered chairs.

四个带编号的椅子。

How many ways can they be seated?

有多少种方法？

To start off, any of the four people can sit

一开始，四个人中的任何一个人

in the first chair.

都可以坐第一把椅子。

One this choice is made,

一旦选定其中一个人

only three people remain standing.

只剩下三个人站着

After the second person sits down,

在第二个人坐下后

only two people are left as candidates

谁坐第三把椅子只有

for the third chair.

两个选择。

And after the third person has sat down,

第三人坐了下来，

the last person standing has no choice

最后一个站的人已别无选择

but to sit in the fourth chair.

只能坐在第四把椅子上。

If we manually write out all the possible arrangements,

如果我们手写出所有可能的安排，

or permutations,

或置换，

it turns out that there are 24 ways

会出现24 种方法

that four people can be seated into four chairs,

让四人可以坐满四把椅子，

but when dealing with larger numbers,

但当处理较大的数字，

this can take quite a while.

这可能会需要相当长的一段时间。

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

所以让我们看看是否有更快的方法。

Going from the beginning again,

我们再一次从头开始

you can see that each of the four initial choices

为第一把椅子

for the first chair

我们有四个初始选项

leads to three more possible choices for the second chair,

这样第二把椅子，我们有三个选项

and each of those choices

每一个选项

leads to two more for the third chair.

使得第三把椅有两个选项

So instead of counting each final scenario individually,

替换费时的累加每一种可能性

we can multiply the number of choices for each chair:

我们可以将每个椅子的可选择数相乘

four times three times two times one

4乘3乘2乘1

to achieve the same result of 24.

得出一样的得数,24。

An interesting pattern emerges.

一个有意思的模式出现了

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

我们从要安排的个体数开始

four in this case,

在这个例子中是四

and multiply it by consecutively smaller integers

然后乘以比这个数小一位的整数

until we reach one.

直到一。

This is an exciting discovery.

这是一个令人兴奋的发现。

So exciting that mathematicians have chosen

数学家们如此兴奋以至于已经决定

to symbolize this kind of calculation,

讲这种据算象征性的取名为

known as a factorial,

阶乘

with an exclamation mark.

并随的一个感叹号。

As a general rule, the factorial of any positive integer

一般规则: 任何正整数的阶乘

is calculated as the product

都是这个整数本身

of that same integer

和每一个比这个整数小的

and all smaller integers down to one.

直到一的整数的乘积。

In our simple example,

在我们的简单示例中，

the number of ways four people

四个人被

can be arranged into chairs

安排坐入椅子的不同可能性

is written as four factorial,

被写作四的阶乘

which equals 24.

这等于 24。

So let's go back to our deck.

所以让我们先前的纸牌例子

Just as there were four factorial ways

正如我们有4种乘积的方法

of arranging four people,

来安排4个人就坐

there are 52 factorial ways

我们有52种阶乘的方法

of arranging 52 cards.

来排列52张牌

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

幸运的是，我们不需要手动计算

Just enter the function into a calculator,

只要把公式输入进计算器

and it will show you that the number of

计算器会告诉你

possible arrangements is

排列的不同方法一共是

8.07 x 10^67,

8.07 x 10 ^67，

or roughly eight followed by 67 zeros.

大约是8后面的67个零。

Just how big is this number?

这个数字有多大？

Well, if a new permutation of 52 cards

如果一种52张牌的排列

were written out every second

用掉1秒钟来写出

starting 13.8 billion years ago,

从138亿年前

when the Big Bang is thought to have occurred,

公认的宇宙大爆炸之时开始

the writing would still be continuing today

我们可以一直写到今天

and for millions of years to come.

并且继续写上数百万年

In fact, there are more possible

事实上，这一副扑克牌的

ways to arrange this simple deck of cards

安排方式要比

than there are atoms on Earth.

地球上原子的数量多。

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

所以在下一次轮到你洗牌时

take a moment to remember

花一点时间来记住

that you're holding something that

你拿着的这副牌

may have never before existed

可能以前并不存在

and may never exist again.

而且可能永远也不会再出现。