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.
抽一張牌,隨便一張, 其實,乾脆把整副牌都攤開來 看一看, 一副共有 52 張的撲克牌, 已沿用了好幾個世紀。 每天,成千上萬副這樣的牌, 在全球各個賭場被洗來洗去, 每次洗都會重新排序。 但當你每回拿起一副洗好的牌, 像這副一樣, 你幾乎可以確定的是, 你手上這副牌的順序 在過去從未出現。 怎麼會這樣? 答案在於,究竟有多少排列組合, 不論是這 52 張牌, 或任何物件, 有多少可能的排列組合存在? 52 看起來不是個很大的數字, 但我們還是先從 更小的數字開始吧。 例如有四個人嘗試坐在 四張有編號的椅子上, 他們的座位有幾種坐法? 一開始,四人中的任何一位 都可以坐在一號位置, 決定好之後, 還有三個人站著, 第二個人坐下之後, 就剩下兩個人有可能 坐在三號位置。 第三個人坐下後, 最後一個站著的人便別無他選, 只能坐在四號椅子。 如果我們寫下 所有可能的座位排法, 或者說排列, (permutations) 結果將有 24 種不同的坐法, 讓四個人坐上四張椅子。 但當要處理的數字較大時, 這就要花上好些時間了。 我們來想想有沒有更快的方法。 從頭來過, 由誰坐上一號椅子, 引出二號椅子的三種可能選擇, 而當中的每個選項, 再引出三號座位的兩種可能性。 所我們不需要 一個一個排出最終的坐法, 只需乘上每張椅子的可能選項: 4 乘以 3 乘以 2 乘以 1。 就會得到相同的結果, 即 24 種坐法。 所以,出現了有趣的規則: 我們先確認要排列的物件數量, 這次是四個人, 然後連續乘以越來越小的整數, 直到 1 為止。 這是很有趣的發現, 數學家將這種計算方法 命名為階乘, 以驚嘆號「!」表示。 一般而言,任意整數的階乘, 計算方法為: 從自己開始,越來越小的整數, 往下相乘,直到 1 為止。 我們剛剛那個簡單的例子, 4 個人座位的排列方法, 就可以寫成 4 的階乘「 4! 」, 計算結果等於 24。 所以讓我們回頭來看這副牌, 如同計算 4 個人 座位的排列方式, 52 張牌就有 52! 種排列方式。 好在我們不需要用手算, 只要按計算機就可以知道, 可能的排列方式共有 8.07 乘以 10 的 67 次方 這麼多種的可能排序, 大約就是 8 後面加上 67 個 0 。 這數字到底是多大呢? 嗯,如果每秒鐘排一種順序, 大約要花 138 億年, 差不多是從 宇宙大爆炸要開始的時候, 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。 事實上,52 張牌的排法, 數量可能遠超過, 地球上所有的原子數目總和。 所以下次輪到你洗牌的時候, 記得想想 你現在洗出來的這副牌, 它的排列順序, 可能是絕無僅有,空前絕後的。