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

1,665,654 views ・ 2014-03-27

TED-Ed


Please double-click on the English subtitles below to play the video.

00:06
Pick a card, any card.
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Actually, just pick up all of them and take a look.
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This standard 52-card deck has been used for centuries.
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Everyday, thousands just like it
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are shuffled in casinos all over the world,
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the order rearranged each time.
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And yet, every time you pick up a well-shuffled deck
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like this one,
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you are almost certainly holding
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an arrangement of cards
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that has never before existed in all of history.
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How can this be?
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The answer lies in how many different arrangements
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of 52 cards, or any objects, are possible.
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Now, 52 may not seem like such a high number,
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but let's start with an even smaller one.
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Say we have four people trying to sit
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in four numbered chairs.
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How many ways can they be seated?
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To start off, any of the four people can sit
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in the first chair.
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One this choice is made,
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only three people remain standing.
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01:01
After the second person sits down,
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only two people are left as candidates
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for the third chair.
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And after the third person has sat down,
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the last person standing has no choice
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but to sit in the fourth chair.
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If we manually write out all the possible arrangements,
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or permutations,
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it turns out that there are 24 ways
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that four people can be seated into four chairs,
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but when dealing with larger numbers,
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this can take quite a while.
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So let's see if there's a quicker way.
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Going from the beginning again,
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you can see that each of the four initial choices
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for the first chair
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leads to three more possible choices for the second chair,
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and each of those choices
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leads to two more for the third chair.
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So instead of counting each final scenario individually,
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we can multiply the number of choices for each chair:
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four times three times two times one
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to achieve the same result of 24.
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An interesting pattern emerges.
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We start with the number of objects we're arranging,
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four in this case,
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and multiply it by consecutively smaller integers
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until we reach one.
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This is an exciting discovery.
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So exciting that mathematicians have chosen
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to symbolize this kind of calculation,
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known as a factorial,
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with an exclamation mark.
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As a general rule, the factorial of any positive integer
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is calculated as the product
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of that same integer
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and all smaller integers down to one.
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In our simple example,
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the number of ways four people
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can be arranged into chairs
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is written as four factorial,
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which equals 24.
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So let's go back to our deck.
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Just as there were four factorial ways
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of arranging four people,
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there are 52 factorial ways
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of arranging 52 cards.
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Fortunately, we don't have to calculate this by hand.
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Just enter the function into a calculator,
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and it will show you that the number of
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possible arrangements is
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8.07 x 10^67,
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or roughly eight followed by 67 zeros.
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Just how big is this number?
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Well, if a new permutation of 52 cards
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were written out every second
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starting 13.8 billion years ago,
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when the Big Bang is thought to have occurred,
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the writing would still be continuing today
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and for millions of years to come.
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In fact, there are more possible
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ways to arrange this simple deck of cards
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than there are atoms on Earth.
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So the next time it's your turn to shuffle,
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take a moment to remember
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that you're holding something that
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may have never before existed
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and may never exist again.
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