I am solving number of arrangements for following question: Eight books are placed on a shelf. Three of them form a 3-volume series, two form a 2-volume series, and 3 stand on their own. In how many ways can the eight books be arranged so that the books in the 3-volume series are placed together according to their correct order, and so are the books in the 2-volume series? Noted that there is only one correct order for each series. I analyzed this problem in following way: 3 volume book as A, 2 volume book as B, and remaining books as C. So three can be arranged as 3!, and C in turn can be arranged as 3! which by product rule we can have 3! * 3! = 36. Why this analysis or approach to this problem is wrong. Correct answer is 120 (i.e. 5!). Kindly help. thanks
Here are the five books: Let's use slots like we did with the license plates: We'll fill each slot -- one at a time... Then we can use the counting principle! The first slot: We have all 5 books to choose from to fill this slot. Let's say we put book C there... Now, we only have 4 books that can go here... How many books are left for this slot? See it? Whoa, dude! That's 5! So, there are 120 ways to arrange five books on a bookshelf. Was the answer to our 3-book problem really 3! ? Yep! Will this always work? TRY IT: How many ways can eight books be arranged on a bookshelf? (reason it out with slots) Page 2
Now, we're going to learn how to count and arrange. (As if just learning to count wasn't exciting enough!) How many ways can we arrange three books on a bookshelf? Here are the books: Well, there's one arrangement. Let's pound out the others: That's all of them... There are 6 ways to arrange three books on a bookshelf. What about five books? Dang! I don't want to have to draw it all out! Let's FIGURE it out instead. Page 3
* For this one, order does NOT matter! We did this problem before: If we have 8 books, how many ways can we arrange 3 on a We figured it out with slots: But, using the formula gave us the same thing: Here's a different question for you: If we have 8 books and we want to take 3 on vacation with us, how What's the difference between these problems? ORDER DOESN'T MATTER! In the first problem, we were arranging the 3 books on a shelf... and in the second problem, we're just tossing the 3 books in a suitcase. So, if order doesn't matter, we'll just divide it out! Arranging the 3 books is 3! Page 4
Grab a calculator! I'm going to teach you about a new button. Look for it... It will either be
(It's probably above one of the other buttons.) Find it? It's called a factorial. Here's an example: (No, this isn't just an excited 5.) Here's what it means: Check it by multiplying it out the long way, then try the button. Here are some others:
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