Question

There are five distinct computer science books, three distinct mathematics books, and two distinct art books. In how many ways can these
books be arranged on a shelf if one of the art books is on the left of all computer science book, and the other art book is on the right of all the computer science books?

Answer 1

didn't you already come up with an answer yourself?

Answer 2

I tried all the possible ways A _ _ _ _ _ _ _ _ A , A _ _ _ _ _ _ _ A _, A _ _ _ _ _ _ A _ _, etc.

Answer 3

no I think it's not correct what I did was

Answer 4

2!*8! + 3!7!+4!6!+5!5! then multipled it by 2 since the two art books can be in reversed order

Answer 5

but I don't think thats the right answer if anyone can help that be great

Answer 6

sorry, I can't think of a solution :/

Answer 7

I think I got it you have to consider how the math books are arrange and art books is just 2 ways and cmpt books are 5!

Answer 8

So let's consider the case where there are no math books between the art books. Then we can consider the art/computer science books as a "chunk". Then, since we have that "chunk" and three math books, we need to think of how many ways that can be arranged -- which is 3! = 6. Next, we determine how many ways in which the math books can be arranged -- only 2. Finally, we consider the number of ways the CS books can be arranged -- which is 5! = 120. Therefore, if there are no math books between the art books, we have 6 * 2 * 120 = 1440 different arrangements.

Answer 9

oops, that should be 4! = 24, so we get 24*2*120 = 5,760

Answer 10

Next we consider the case where we have one math book inside the art books. There are three possible choices for which math book goes inside the art books, so we get a factor of 3. Next, we have our "chunk" and 2 other books, which can be arranged in 3! = 6 different ways. Our art books can be arranged in 2 ways, as before. We have 6 books between the art books, which can be arranged in 6! = 720 different ways. All of this together gives 3 * 6 * 2 *720 = 25,920 combinations.

Answer 11

Then we consider the case where we have two math books inside the art books. There are three possible choices of which two math books go inside, giving us a factor 3. We have our "chunk" plus one other math book, which can be arranged in 2 ways. Our art books can be arranged in 2 ways, and the books in between can be arranged in 7! = 5040 ways. All of this together yields 3 * 2 * 2 * 5040 = 60,480 different arrangements.

Answer 12

Lastly we consider the case where we have all three math books inside the art books. Clearly now, our art books can be arranged in 2 ways, and the others can be arranged in 8! = 40,320 ways, yielding a total number of arrangements of 80,640. Summing all of these together, we have 5760 + 25920 + 60480 + 80640 = 172,800 different possible arrangements for our books. Whew.