probability problem: There are ten books on the shelf...

[krzesimort]

Hi, I have a math problem which I've been trying to solve but I couldn't invent a solution. In the past I did only simple problems related to probability so this one is really hard for me. Here is the problem:

There are ten books on the shelf. 5 of them are about cars, 3 of them are about animals and 2 of them are about plants. If we know that the first one is about cars, the seventh is about plants, the last one is about animals and the rest of the books are placed randomly - what is the probability that books that are about the same thing are next to each other?

I would be really grateful for help.

 

[pka]

Because these are books it is reasonable to assume that they are distinct.
There are [imath](5)\cdot(2)\cdot(3)[/imath] ways to choose the first, seventh, & last books.
That leaves seven books the be places in any order. How many is that? Now multiply.

 

[krzesimort]

I understand the first part so we have seven books to be placed, however, what exactly should I do to calculate the probability I mentioned in my question?

 

[Dr.Peterson]

There are ten books on the shelf. 5 of them are about cars, 3 of them are about animals and 2 of them are about plants. If we know that the first one is about cars, the seventh is about plants, the last one is about animals and the rest of the books are placed randomly - what is the probability that books that are about the same thing are next to each other?

I always start by trying to be sure what the problem means. And I have a problem here: Exactly what does it mean that "books that are about the same thing are next to each other"? Does it mean that some one pair of books on the same topic are together, or, more likely, that all books on each topic are adjacent, or perhaps something else?

Here are the books, with all books on a topic together (I'm assuming, like pka, that the books are distinguishable, but not denoting that here):

C C C C C A A A P P​

Here are the books whose locations are known:

C _ _ _ _ _ P _ _ A​

How many ways are there to fill the remaining 7 places with the 7 remaining books, with no condition?

If I'm right that "success" means that all C's, all A's, and all P's are together, the only successful arrangements look like this:

C C C C C P P A A A​

How many ways are there to arrange the 5 remaining C's, the one remaining P, and the two remaining A's in these positions?

It will also help us if you can tell us more specifically what you know and where you need help. If this is for a class, what topics have you learned? If not, where does it come from, and what topics have you learned? And can you show us the hardest problem you have solved, and how? All of that will help us know what we can use to help.