1 00:00:06,954 --> 00:00:09,124 Pick a card, any card. 2 00:00:09,124 --> 00:00:12,014 Actually, just pick up all of them and take a look. 3 00:00:12,014 --> 00:00:15,848 This standard 52 card deck has been used for centuries. 4 00:00:15,848 --> 00:00:18,098 Everyday, thousands just like it 5 00:00:18,098 --> 00:00:21,134 are shuffled in casinos all over the world, 6 00:00:21,134 --> 00:00:23,719 the order rearranged each time. 7 00:00:23,719 --> 00:00:26,431 And yet, every time you pick up a well shuffled deck 8 00:00:26,431 --> 00:00:27,642 like this one, 9 00:00:27,642 --> 00:00:29,431 you are almost certainly holding 10 00:00:29,431 --> 00:00:30,848 an arrangement of cards 11 00:00:30,848 --> 00:00:33,729 that has never before existed in all of history. 12 00:00:33,729 --> 00:00:35,764 How can this be? 13 00:00:35,764 --> 00:00:37,900 The answer lies in how many different arrangements 14 00:00:37,900 --> 00:00:42,348 of 52 cards, or any objects, are possible. 15 00:00:42,348 --> 00:00:45,620 Now, 52 may not seem like such a high number, 16 00:00:45,620 --> 00:00:48,035 but let's start with an even smaller one. 17 00:00:48,035 --> 00:00:49,932 Say we have four people trying to sit 18 00:00:49,932 --> 00:00:52,348 in four numbered chairs. 19 00:00:52,348 --> 00:00:54,460 How many ways can they be seated? 20 00:00:54,460 --> 00:00:56,598 To start off, any of the four people can sit 21 00:00:56,598 --> 00:00:57,920 in the first chair. 22 00:00:57,920 --> 00:00:59,132 One this choice is made, 23 00:00:59,132 --> 00:01:01,466 only three people remain standing. 24 00:01:01,466 --> 00:01:03,262 After the second person sits down, 25 00:01:03,262 --> 00:01:05,219 only two people are left as candidates 26 00:01:05,219 --> 00:01:06,680 for the third chair. 27 00:01:06,680 --> 00:01:08,680 And after the third person has sat down, 28 00:01:08,680 --> 00:01:10,431 the last person standing has no choice 29 00:01:10,431 --> 00:01:12,347 but to sit in the fourth chair. 30 00:01:12,347 --> 00:01:15,098 If we manually write out all the possible arrangements, 31 00:01:15,098 --> 00:01:16,814 or permutations, 32 00:01:16,814 --> 00:01:18,818 it turns out that there are 24 ways 33 00:01:18,818 --> 00:01:22,180 that four people can be seated in to four chairs, 34 00:01:22,180 --> 00:01:23,991 but when dealing with larger numbers, 35 00:01:23,991 --> 00:01:25,532 this can take quite a while. 36 00:01:25,532 --> 00:01:27,848 So let's see if there's a quicker way. 37 00:01:27,848 --> 00:01:29,286 Going from the beginning again, 38 00:01:29,286 --> 00:01:31,370 you can see that each of the four initial choices 39 00:01:31,370 --> 00:01:32,682 for the first chair 40 00:01:32,682 --> 00:01:35,999 leads to three more possible choices for the second chair, 41 00:01:35,999 --> 00:01:37,461 and each of those choices 42 00:01:37,461 --> 00:01:39,847 leads to two more for the third chair. 43 00:01:39,847 --> 00:01:43,181 So instead of counting each final scenario individually, 44 00:01:43,181 --> 00:01:46,262 we can multiply the number of choices for each chair: 45 00:01:46,262 --> 00:01:49,096 four times three times two times one 46 00:01:49,096 --> 00:01:51,848 to achieve the same result of 24. 47 00:01:51,848 --> 00:01:53,681 An interesting pattern emerges. 48 00:01:53,681 --> 00:01:56,729 We start with the number of objects we're arranging, 49 00:01:56,729 --> 00:01:58,098 four in this case, 50 00:01:58,098 --> 00:02:00,847 and multiply it by consecutively smaller integers 51 00:02:00,847 --> 00:02:02,902 until we reach one. 52 00:02:02,902 --> 00:02:04,514 This is an exciting discovery. 53 00:02:04,514 --> 00:02:06,449 So exciting that mathematicians have chosen 54 00:02:06,449 --> 00:02:08,575 to symbolize this kind of calculation, 55 00:02:08,575 --> 00:02:10,345 known as a factorial, 56 00:02:10,345 --> 00:02:12,038 with an exclamation mark. 57 00:02:12,038 --> 00:02:15,514 As a general rule, the factorial of any positive integer 58 00:02:15,514 --> 00:02:17,416 is calculated as the product 59 00:02:17,416 --> 00:02:18,876 of that same integer 60 00:02:18,876 --> 00:02:21,836 and all smaller integers down to one. 61 00:02:21,836 --> 00:02:23,263 In our simple example, 62 00:02:23,263 --> 00:02:24,596 the number of ways four people 63 00:02:24,596 --> 00:02:26,181 can be arranged in to chairs 64 00:02:26,181 --> 00:02:28,052 is written as four factorial, 65 00:02:28,052 --> 00:02:29,975 which equals 24. 66 00:02:29,975 --> 00:02:31,808 So let's go back to our deck. 67 00:02:31,808 --> 00:02:33,598 Just as there were four factorial ways 68 00:02:33,598 --> 00:02:35,431 of arranging four people, 69 00:02:35,431 --> 00:02:37,598 there are 52 factorial ways 70 00:02:37,598 --> 00:02:40,014 of arranging 52 cards. 71 00:02:40,014 --> 00:02:43,066 Fortunately, we don't have to calculate this by hand. 72 00:02:43,066 --> 00:02:45,014 Just enter the function in to a calculator 73 00:02:45,014 --> 00:02:46,431 and it will show you that the number of 74 00:02:46,431 --> 00:02:47,931 possible arrangements is 75 00:02:47,931 --> 00:02:52,368 8.07 x 10^67, 76 00:02:52,368 --> 00:02:55,788 or roughly eight followed by 67 zeros. 77 00:02:55,788 --> 00:02:57,458 Just how big is this number? 78 00:02:57,458 --> 00:02:59,708 Well, if a new permutation of 52 cards 79 00:02:59,708 --> 00:03:01,752 were written out every second 80 00:03:01,752 --> 00:03:04,378 starting 13.8 billion years ago, 81 00:03:04,378 --> 00:03:06,344 when the big bang is thought to have occurred, 82 00:03:06,344 --> 00:03:09,094 the writing would still be continuing today 83 00:03:09,094 --> 00:03:11,676 and for millions of years to come. 84 00:03:11,676 --> 00:03:13,426 In fact, there are more possible 85 00:03:13,426 --> 00:03:16,345 ways to arrange this simple deck of cards 86 00:03:16,345 --> 00:03:18,593 than there are atoms on Earth. 87 00:03:18,593 --> 00:03:20,759 So the next time it's your turn to shuffle, 88 00:03:20,759 --> 00:03:22,093 take a moment to remember 89 00:03:22,093 --> 00:03:23,174 that you're holding something that 90 00:03:23,174 --> 00:03:25,235 may have never before existed 91 00:03:25,235 --> 00:03:27,344 and may never exist again.