Suppose that I have an unlimited supply of identical math books, history books, and physics books. All are the same size, and I have room on a shelf for 8 books. In how many ways can I arrange eight books on the shelf if no two books of the same type can be adjacent?

There 384 ways.

Step-by-step explanation:

We can calculate the number of ways using a rule of multiplication as:

3 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 384

1st Book 2nd 3rd 4th 5th 6th 7th 8th

Because we have space for 8 books. Additionally, the first book can be any of the three types of books: Maths, History or physics. it means that we have 3 options for the 1st book.

But the options of the following spaces are limited by the book that is beside. So, if, for example, the first book is a Math Book, the second book just has 2 options: History or physics and the same situation apply for the 3rd to 8th book on the shelf.

So, there are 384 ways to arrange 8 books on the shelf.