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.
Izaberite kartu. Bilo koju. U stvari, uzmite ih sve i pogledajte. Ovaj standardni špil od 52 karte koristi se vekovima. Svakodnevno, hiljade ovakvih se meša u kazinima širom sveta i redosled karata se menja svaki put. Svaki put kad uzmete dobro promešan špil kao što je ovaj, skoro sigurno ćete imati raspored karata koji nikada u istoriji nije postojao. Kako je to moguće? Odgovor leži u tome koliko ima mogućih različitih rasporeda 52 karte, ili bilo kojih drugih predmeta. Možda 52 ne izgleda kao naročito veliki broj, ali hajde da krenemo sa još manjim. Recimo da imamo četvoro ljudi koji pokušavaju da sednu na četiri numerisane stolice. Na koliko načina mogu da se rasporede? Za početak, svako od njih četvoro može da sedne na prvu stolicu. Kad je ovaj izbor napravljen, samo troje ostaje da stoji. Pošto druga osoba sedne, ostaje samo dva kandidata za treću stolicu. A kad treća osoba sedne, poslednja koja je ostala nema drugog izbora, nego da sedne na četvrtu stolicu. Ako ručno napišemo sve moguće rasporede, ili permutacije, ispostavlja se da postoji 24 načina da se četvoro ljudi rasporedi na 4 stolice, ali kada radimo sa većim brojevima, to može da potraje. Pa, da vidimo da li postoji brži način. Ako krenemo opet od početka, možete videti da svaka od prvobitne 4 mogućnosti za prvu stolicu vodi do još tri mogućnosti za drugu stolicu, a svaka od tih mogućnosti vodi do još dve za treću stolicu. Pa umesto brojanja svakog pojedinačnog rezultata, možemo pomnožiti broj mogućnosti za svaku stolicu: 4 x 3 x 2 x 1, da bismo dobili isti rezultat: 24. Pojavljuje se zanimljiv obrazac. Počinjemo sa brojem predmeta koje raspoređujemo, u ovom slučaju četiri, i množimo ga sledećim manjim celim brojevima dok ne stignemo do 1. Ovo je bilo uzbudljivo otkriće, do te mere, da su matematičari odlučili da predstave ovu operaciju poznatu kao faktorijel, simbolom uzvičnika. Kao opšte pravilo, faktorijel bilo kog pozitivnog celog broja se računa kao proizvod tog istog celog broja i svih manjih celih brojeva od njega, sve do broja 1. U našem jednostavnom primeru, broj načina na koje se četvoro ljudi može rasporediti na stolice je napisan kao četiri faktorijel, što iznosi 24. Da se vratimo na naš špil. Isto kao što je bilo četiri faktorijel načina raspoređivanja četvoro ljudi, tako postoji 52 faktorijel načina da se rasporede 52 karte. Srećom, ne moramo to da računamo ručno. Samo upišite funkciju u digitron i on će vam pokazati da je broj mogućih rasporeda 8.07 x 10^67, što je otprilike - broj 8 sa 67 nula. Koliki je ustvari ovaj broj? Pa, ako bi se svaka nova permutacija 52 karte zapisivala svake sekunde počevši od pre 13,8 milijardi godina, kada se veruje da se dogodio Veliki prasak, zapisivanje bi trajalo i danas i nastavilo bi se još milionima godina. U stvari, ima više mogućih načina rasporeda ovog jednostavnog špila karata, nego što ima atoma na Zemlji. Zato, sledeći put kad bude bio vaš red da mešate, setite se da možda držite nešto što nikada ranije nije postojalo i neće ni postojati.