Combinatorics Problem

**Sarah200488** · March 29th 2008, 09:54 PM

If you have five biology books and three English books (assume that they are NOT all identical- each book is different), how many ways can you arrange them on a shelf so that no more than 2 books of any subject are placed together?

For an assignment due tomorrow- I've tried every other option!

Thank you!!

**Plato** · March 30th 2008, 05:11 AM

Originally Posted by Sarah200488

If you have five biology books and three English books (assume that they are NOT all identical- each book is different), how many ways can you arrange them on a shelf so that no more than 2 books of any subject are placed together?

This can be a bit long. Suppose the E is the event that all English texts are together, B is the event that three of the biology texts are together, and T is the total.
Now your answer is $\left| T \right| - \left| {E \cup B} \right| = \left| T \right| - \left[ {\left| E \right| + \left| B \right| - \left| {E \cap B} \right|} \right]$ where |A| is the number in event A.
|T|=(8!) and $\left| B \right| = {\binom {5}{3}}\left( {3!} \right)\left( {6!} \right)$ .
Can you explain those two and finish?

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