Question 265871: Question: I have 12 textbooks to arrange on a bookshelf. 5 of them are English textbooks and I would like to arrange the books so that at least two of the English textbooks are adjacent to one another. How many different arrangements are possible?
This is how far I have gotten:
(# of ways that all are touching) - (# of ways that none are touching)
(5!)(7!)(2!) - ????
If I am on the right track... how do I figure out the (# of ways that none are touching)?
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Question: I have 12 textbooks to arrange on a bookshelf. 5 of them are English textbooks and I would like to arrange the books so that at least two of the English textbooks are adjacent to one another. How many different arrangements are possible?
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Answer: total # of arrangements - # of arrangements with no two english adjacent
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# no 2 are adjacent = ?
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Four non-englsh will be required to separate the 5 english.
# of ways to select the 4 separaters: 7C4 = 35
Arrangements of the 5 english: 5!
Arrangements of the 4 separaters: 4!
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That leaves 3 books to arrange in 3! ways
Total arrangements: 5!*35*4!*3* = 604,800
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So, # no 2 are adjacent = 604800
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Finally, # at least 2 are adjacent)= 12! - 604800 = 478,396,800
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Note: Interesting problem. Let me know when you can
confirm an answer.
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Cheers,
Stan H.
This is how far I have gotten:
(# of ways that all are touching) - (# of ways that none are touching)
(5!)(7!)(2!) - ????
If I am on the right track... how do I figure out the (# of ways that none are touching)?
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