ClassX Video Lessons Subjects Algebra How many ways can you arrange a deck of cards? – Yannay Khaikin
Imagine picking up a deck of cards. This standard 52-card deck has been around for centuries and is shuffled countless times every day in casinos worldwide. But here’s something amazing: every time you shuffle a deck well, you’re probably holding a unique arrangement of cards that has never existed before in history. How is that possible? The answer lies in the number of different ways you can arrange 52 cards, or any set of objects.
Let’s start with a smaller example. Suppose you have four people who need to sit in four numbered chairs. How many ways can they sit? To figure this out, think about it step by step. Any of the four people can sit in the first chair. Once someone sits down, only three people are left to choose from for the second chair. After the second person sits, only two people remain for the third chair. Finally, the last person has no choice but to sit in the fourth chair.
If you list all the possible seating arrangements, you’ll find there are 24 different ways to seat four people in four chairs. But writing them all out takes time, especially with larger numbers. So, is there a faster way? Yes! You can multiply the number of choices for each chair: four choices for the first chair, three for the second, two for the third, and one for the last. Multiply these together: 4 x 3 x 2 x 1, which equals 24.
There’s a cool pattern here. You start with the number of objects you’re arranging, which is four in this case, and multiply it by smaller numbers until you reach one. This calculation is called a factorial, and it’s so special that mathematicians use an exclamation mark to represent it. The factorial of any positive number is the product of that number and all smaller numbers down to one. For example, the number of ways to arrange four people is written as 4! (four factorial), which equals 24.
Now, let’s return to our deck of cards. Just like there are 4! ways to arrange four people, there are 52! (52 factorial) ways to arrange 52 cards. Luckily, we don’t have to do this calculation by hand. A calculator can show you that 52! is approximately 8.07 x 1067, which is a number with 67 zeros after it!
How big is this number? Imagine writing out a new arrangement of 52 cards every second since the Big Bang, about 13.8 billion years ago. You’d still be writing today and for millions of years more! In fact, there are more ways to arrange a deck of cards than there are atoms on Earth. So, next time you shuffle a deck, remember that you’re holding something truly unique, something that may have never existed before and might never exist again.
Grab a deck of cards and shuffle it thoroughly. Then, try to arrange the cards in a specific order, such as all hearts in sequence from Ace to King. Time yourself to see how quickly you can achieve this. Discuss with your classmates how the concept of factorials applies to the number of possible arrangements you could have made.
Work in teams to solve factorial problems as quickly as possible. Each team member must calculate the factorial of a different number, starting from 1! up to 5!. The first team to correctly complete all calculations wins. Reflect on how the factorial concept helps in understanding permutations and arrangements.
Simulate the seating arrangement problem with your classmates. Use chairs and assign each student a number. Experiment with different seating orders and calculate the total number of possible arrangements using factorials. Discuss how this activity relates to arranging objects in different orders.
Shuffle a deck of cards and record the order of cards. Compare your arrangement with those of your classmates. Discuss why it’s likely that each of you has a unique arrangement, using the concept of 52 factorial to explain the vast number of possibilities.
Create a visual representation of factorials using art supplies. For example, use colored paper to represent different numbers and arrange them in factorial order. Display your artwork and explain how it illustrates the concept of factorials and permutations.
Here’s a sanitized version of the provided YouTube transcript:
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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. Every day, 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. Once 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 approximately 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.
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This version maintains the original content while ensuring clarity and coherence.
Arrangements – Different orders or positions in which a set of items can be organized. – In how many arrangements can the letters in the word “MATH” be placed?
Factorial – The product of all positive integers up to a given number, denoted by the symbol “!” – To find the number of ways to arrange 5 books on a shelf, calculate 5 factorial, which is 5! = 120.
Multiply – To find the product by combining equal groups or adding a number to itself a specified number of times. – To find the area of a rectangle, you multiply its length by its width.
Unique – Being the only one of its kind; distinct or different from others. – Each solution to the equation has a unique value for the variable.
Objects – Items or things that can be counted or measured in a mathematical context. – When arranging objects in a line, the order matters.
Choices – Different options or possibilities available in a situation. – When selecting toppings for a pizza, you have several choices.
Number – A mathematical object used to count, measure, and label. – The number of students in the class is 30.
Cards – Flat, rectangular pieces used in games, often numbered or marked with symbols. – In a standard deck, there are 52 cards.
Ways – Different methods or approaches to achieve a result or solve a problem. – There are several ways to solve a quadratic equation.
Chairs – Seats used for sitting, often used in problems involving arrangements or combinations. – If there are 5 chairs and 3 people, in how many ways can the people be seated?
