Pick a card, any card. Actually, just pick up all of them and take a look.
选一张牌，任何牌。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 are shuffled in casinos all over the world, the order rearranged each time.
每天成千上万像这样的扑克牌在世界各地的赌场中洗牌，每一次排列组合都会改变。
And yet, every time you pick up a well-shuffled deck like this one,
事实上，每一次你从洗过的牌堆里抽一张牌，像这样，
you are almost certainly holding an arrangement of cards that has never before existed in all of history. How can this be?
几乎可以肯定你拥有的牌的排列组合顺序在历史上从未出现过。为什么是这样？
The answer lies in how many different arrangements of 52 cards, or any objects, are possible.
藏在这52张牌的答案有许多可能的排列组合。
Now, 52 may not seem like such a high number, but let's start with an even smaller one.
现在，52看起并不是一个大数字，让我们从一个更小的数字开始研究。
Say we have four people trying to sit in four numbered chairs. How many ways can they be seated?
假设有四个人要坐四个带编号的椅子。有多少种方法？
To start off, any of the four people can sit in the first chair.
一开始，四个人中的任何一个人都可以坐第一把椅子。
One this choice is made, only three people remain standing.
一旦选定其中一个人，只剩下三个人站着。
After the second person sits down, only two people are left as candidates for the third chair.
在第二个人坐下后，谁坐第三把椅子只有两个选择。
And after the third person has sat down, the last person standing has no choice but to sit in the fourth chair.
第三人坐了下来，最后一个站的人已别无选择，只能坐在第四把椅子上。
If we manually write out all the possible arrangements, or permutations,
如果我们手写出所有可能的排列，或置换，
it turns out that there are 24 ways that four people can be seated into four chairs,
会出现24种让四人可以坐满四把椅子的方法，
but when dealing with larger numbers, this can take quite a while.
但当处理较大的数字，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, you can see that each of the four initial choices for the first chair
我们再一次从头开始，第一把椅子，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 leads to two more for the third chair.
每一个选项使得第三把椅有两个选项。
So instead of counting each final scenario individually, we can multiply the number of choices for each chair:
我们不需要一个一个排出最终的坐法，只需乘上每张椅子的可能选项：
four times three times two times one to achieve the same result of 24.
4乘3乘2乘1，得出一样的结果，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, four in this case,
我们从要排列的个体数开始，在这个例子中是四，
and multiply it by consecutively smaller integers until we reach one. This is an exciting discovery.
然后乘以比这个数小一位的整数，直到一。这是一个令人兴奋的发现。
So exciting that mathematicians have chosen to symbolize this kind of calculation, known as a factorial, with an exclamation mark.
数学家们如此兴奋，以至于已经决定将这种计算方法取名为阶乘，并用一个感叹号表示。
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.
一般规则: 任何正整数的阶乘都是这个整数本身和每一个比这个整数小的直到一的整数的乘积。
In our simple example, the number of ways four people can be arranged into chairs is written as four factorial, which equals 24.
在我们的简单示例中，四个人被安排坐入椅子的不同可能性被写作四的阶乘，这等于24。
So let's go back to our deck.
所以让我们回到先前的纸牌例子。
Just as there were four factorial ways of arranging four people, there are 52 factorial ways of arranging 52 cards.
正如我们有4种乘积的方法来安排4个人就坐，我们有52种阶乘的方法来排列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, and it will show you that the number of possible arrangements is 8.07x10^67,
只要把公式输入进计算器，计算器会告诉你排列的不同方法一共是8.07x10^67，
or roughly eight followed by 67 zeros. Just how big is this number?
大约是8后面跟67个零。这个数字有多大？
Well, if a new permutation of 52 cards were written out every second
如果一种52张牌的排列要用1秒钟来写，
starting 13.8 billion years ago, when the Big Bang is thought to have occurred,
那么从138亿年前公认的宇宙大爆炸之时开始，
the writing would still be continuing today and for millions of years to come.
我们可以一直写到今天，并且继续写上数百万年。
In fact, there are more possible ways to arrange this simple deck of cards than there are atoms on Earth.
事实上，这一副扑克牌的排列方式要比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 that you're holding something that may have never before existed and may never exist again.
花一点时间来记住你拿着的这副牌可能以前并不存在，而且可能永远也不会再出现。