How many ways can you arrange a deck of cards? - Yannay Khaikin

Na koliko načina možete da poređate špil karata? - Janaj Kajkin (Yannay Khaikin)

Prevodilac: Miloš Milosavljević Lektor: Mile Živković

Pick a card, any card.

Izaberite kartu. Bilo koju.

Actually, just pick up all of them and take a look.

U stvari, uzmite ih sve i pogledajte.

This standard 52-card deck has been used for centuries.

Ovaj standardni špil od 52 karte koristi se vekovima.

Everyday, thousands just like it

Svakodnevno, hiljade ovakvih

are shuffled in casinos all over the world,

se meša u kazinima širom sveta

the order rearranged each time.

i redosled karata se menja svaki put.

And yet, every time you pick up a well-shuffled deck

Svaki put kad uzmete dobro promešan špil

like this one,

kao što je ovaj,

you are almost certainly holding

skoro sigurno ćete imati

an arrangement of cards

raspored karata

that has never before existed in all of history.

koji nikada u istoriji nije postojao.

How can this be?

Kako je to moguće?

The answer lies in how many different arrangements

Odgovor leži u tome koliko ima mogućih različitih rasporeda

of 52 cards, or any objects, are possible.

52 karte, ili bilo kojih drugih predmeta.

Now, 52 may not seem like such a high number,

Možda 52 ne izgleda kao naročito veliki broj,

but let's start with an even smaller one.

ali hajde da krenemo sa još manjim.

Say we have four people trying to sit

Recimo da imamo četvoro ljudi koji pokušavaju da sednu

in four numbered chairs.

na četiri numerisane stolice.

How many ways can they be seated?

Na koliko načina mogu da se rasporede?

To start off, any of the four people can sit

Za početak, svako od njih četvoro može da sedne

in the first chair.

na prvu stolicu.

One this choice is made,

Kad je ovaj izbor napravljen,

only three people remain standing.

samo troje ostaje da stoji.

After the second person sits down,

Pošto druga osoba sedne,

only two people are left as candidates

ostaje samo dva kandidata

for the third chair.

za treću stolicu.

And after the third person has sat down,

A kad treća osoba sedne,

the last person standing has no choice

poslednja koja je ostala nema drugog izbora,

but to sit in the fourth chair.

nego da sedne na četvrtu stolicu.

If we manually write out all the possible arrangements,

Ako ručno napišemo sve moguće rasporede,

or permutations,

ili permutacije,

it turns out that there are 24 ways

ispostavlja se da postoji 24 načina

that four people can be seated into four chairs,

da se četvoro ljudi rasporedi na 4 stolice,

but when dealing with larger numbers,

ali kada radimo sa većim brojevima,

this can take quite a while.

to može da potraje.

So let's see if there's a quicker way.

Pa, da vidimo da li postoji brži način.

Going from the beginning again,

Ako krenemo opet od početka,

you can see that each of the four initial choices

možete videti da svaka od prvobitne 4 mogućnosti

for the first chair

za prvu stolicu

leads to three more possible choices for the second chair,

vodi do još tri mogućnosti za drugu stolicu,

and each of those choices

a svaka od tih mogućnosti

leads to two more for the third chair.

vodi do još dve za treću stolicu.

So instead of counting each final scenario individually,

Pa umesto brojanja svakog pojedinačnog rezultata,

we can multiply the number of choices for each chair:

možemo pomnožiti broj mogućnosti za svaku stolicu:

four times three times two times one

4 x 3 x 2 x 1,

to achieve the same result of 24.

da bismo dobili isti rezultat: 24.

An interesting pattern emerges.

Pojavljuje se zanimljiv obrazac.

We start with the number of objects we're arranging,

Počinjemo sa brojem predmeta koje raspoređujemo,

four in this case,

u ovom slučaju četiri,

and multiply it by consecutively smaller integers

i množimo ga sledećim manjim celim brojevima

until we reach one.

dok ne stignemo do 1.

This is an exciting discovery.

Ovo je bilo uzbudljivo otkriće,

So exciting that mathematicians have chosen

do te mere, da su matematičari odlučili

to symbolize this kind of calculation,

da predstave ovu operaciju

known as a factorial,

poznatu kao faktorijel,

with an exclamation mark.

simbolom uzvičnika.

As a general rule, the factorial of any positive integer

Kao opšte pravilo, faktorijel bilo kog pozitivnog celog broja

is calculated as the product

se računa kao proizvod

of that same integer

tog istog celog broja

and all smaller integers down to one.

i svih manjih celih brojeva od njega, sve do broja 1.

In our simple example,

U našem jednostavnom primeru,

the number of ways four people

broj načina na koje se četvoro ljudi

can be arranged into chairs

može rasporediti na stolice

is written as four factorial,

je napisan kao četiri faktorijel,

which equals 24.

što iznosi 24.

So let's go back to our deck.

Da se vratimo na naš špil.

Just as there were four factorial ways

Isto kao što je bilo četiri faktorijel načina

of arranging four people,

raspoređivanja četvoro ljudi,

there are 52 factorial ways

tako postoji 52 faktorijel načina

of arranging 52 cards.

da se rasporede 52 karte.

Fortunately, we don't have to calculate this by hand.

Srećom, ne moramo to da računamo ručno.

Just enter the function into a calculator,

Samo upišite funkciju u digitron

and it will show you that the number of

i on će vam pokazati da je

possible arrangements is

broj mogućih rasporeda

8.07 x 10^67,

8.07 x 10^67,

or roughly eight followed by 67 zeros.

što je otprilike - broj 8 sa 67 nula.

Just how big is this number?

Koliki je ustvari ovaj broj?

Well, if a new permutation of 52 cards

Pa, ako bi se svaka nova permutacija 52 karte

were written out every second

zapisivala svake sekunde

starting 13.8 billion years ago,

počevši od pre 13,8 milijardi godina,

when the Big Bang is thought to have occurred,

kada se veruje da se dogodio Veliki prasak,

the writing would still be continuing today

zapisivanje bi trajalo i danas

and for millions of years to come.

i nastavilo bi se još milionima godina.

In fact, there are more possible

U stvari, ima više

ways to arrange this simple deck of cards

mogućih načina rasporeda ovog jednostavnog špila karata,

than there are atoms on Earth.

nego što ima atoma na Zemlji.

So the next time it's your turn to shuffle,

Zato, sledeći put kad bude bio vaš red da mešate,

take a moment to remember

setite se da

that you're holding something that

možda držite nešto

may have never before existed

što nikada ranije nije postojalo

and may never exist again.

i neće ni postojati.