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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In how many ways can 5 different books be arranged on a shelf if 2 books are never together Since 2 books are never together, we can arrange these two books at 4 places (2 places in between the remaining 3 books + 2 at the ends) in 4P2 ways. After this the remaining 3 books can be arranged in 3P3 ways. Hence, the total number of arrangements in which 2 books are never together = 4P2 × 3P3 = `(4!)/(2!) xx 3!` = `(4 xx 3 xx 2 xx 1)/(2!) xx 3 xx 2!` = 72. Concept: Permutations - Permutations When Repetitions Are Allowed Is there an error in this question or solution? |