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 an 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 into 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 kind 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 into 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 into 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? Well, if a new permutation of 52 cards were written out every second starting 13.8 billion years ago, when the Big Bang is thought to have occurred, the writing would still be continuing today and for millions of years to come. In fact, there are more possible ways to arrange this simple deck of cards than there are atoms on Earth. So the next time it's your turn to shuffle, take a moment to remember that you're holding something that may have never before existed and may never exist again.
选一张牌,任何牌。 事实上,把它们全部拿起来看一看 标准的 52 张牌已经延用了几个世纪之久。 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 事实上, 每一次你从洗过的牌堆里抽一张牌 像这样, 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? 答案藏在这52张牌有 许多可能的排列组合 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。 假设有四个人要坐 四个带编号的椅子。 有多少种方法? 一开始,四个人中的任何一个人 都可以坐第一把椅子。 一旦选定其中一个人 只剩下三个人站着 在第二个人坐下后 谁坐第三把椅子只有 两个选择。 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。 如果我们手写出所有可能的安排, 或置换, 会出现24 种方法 让四人可以坐满四把椅子, 但当处理较大的数字, 这可能会需要相当长的一段时间。 所以让我们看看是否有更快的方法。 我们再一次从头开始 为第一把椅子 我们有四个初始选项 这样第二把椅子,我们有三个选项 每一个选项 使得第三把椅有两个选项 替换费时的累加每一种可能性 我们可以将每个椅子的可选择数相乘 4乘3乘2乘1 得出一样的得数,24。 一个有意思的模式出现了 我们从要安排的个体数开始 在这个例子中是四 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。 数学家们如此兴奋以至于已经决定 讲这种据算象征性的取名为 阶乘 并随的一个感叹号。 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。 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。 所以让我们先前的纸牌例子 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 幸运的是,我们不需要手动计算 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 8.07 x 10 ^67, 大约是8后面的67个零。 这个数字有多大? 如果一种52张牌的排列 用掉1秒钟来写出 从138亿年前 公认的宇宙大爆炸之时开始 我们可以一直写到今天 并且继续写上数百万年 事实上,这一副扑克牌的 安排方式要比 地球上原子的数量多。 所以在下一次轮到你洗牌时 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。