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.
(Aren't you glad I didn't make you draw them out?)
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)
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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.
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* 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
bookshelf?
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
many ways can we do it?
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!
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Grab a calculator! I'm going to teach you about a new button.
Look for it... It will either be
| or |
(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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