How many options there are to choose 5 books from 12 different books, that the books are not neighbors. There are 2 cases: 1. The order of the choosing is matter. 2. The order doesn't matter.
I really don't know where to start.
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5Write A for books you choose and B for books you don't. Then the 12 books look like this: ABABABABA, with another three B's inserted somewhere. – user49640 Jun 14 '17 at 20:45
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Then to follow on from @user49640 's great hint, figure out the number of ways that you can put the other three B's, and you're well on your way. – John Jun 14 '17 at 20:53
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Is it (3 from 10)? But the Bs could be different eaxh tine – Fayubla Jun 14 '17 at 20:57
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If you really don't know, it won't be too long to write down all the possibilities. While you're doing that you'll start to get ideas about how you can do it systematically. I'm sure that's how 99% of us learned to do these things originally. How many slots do you have in which to put the extra B's? – user49640 Jun 14 '17 at 21:05
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If I can see the way I will understand it – Fayubla Jun 15 '17 at 04:10
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And I have 6 more slots for the Bs – Fayubla Jun 15 '17 at 04:11
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6^3 (3 Bs left) – Fayubla Jun 15 '17 at 04:12
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You have six slots for the Bs. That's correct. But all that matters for determing where the A's are located is how many B's you put in each slot, not which B's. So the answer is not $6^3$. – user49640 Jun 15 '17 at 23:58
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(6-1+3) choose (6-1) = (8!)/(5!*3!) – Fayubla Jun 16 '17 at 03:47
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Yes, that works. – user49640 Jun 20 '17 at 06:20