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4 math books, 3 physics books, and 2 history books are to be placed on a shelf. The math books must be kept together and must be arranged in a particular order because they are volumes 1-4 of a series. The physics books are to be kept together, but not necessarily in any particular order. The history books can go anywhere. How many ways are there in which the books can be arranged and still meet these criteria?

On the highest level we are ordering 4 objects: (1) the math books (2) the physics books (3) first history book (4) second history book. No physics or history books are allowed to enter the box containing the math books. Similarly no math or history books are allowed to enter the box containing the physics books.

There are 4! = 24 ways to arrange these four objects.

The math books can only be kept in one order. Permuting the order of the math books is not allowed. So there’s only 1 or 1! ways to arrange the math books.

The physics books can be permuted to be in any order so long as they are kept together. There are 3 physics books, so there are 3! = 6 ways to arrange them.

All these things are happening simultaneously, therefore we multiply them all together to get

4! x 1! x 3! = 24 x 1 x 6 = 144 possible arrangements.