### Instruction: How many ways can we put 3 math books and 5 English books on a shelf if all the math books must stay together and all the English books must also stay together?  (The math books are all different and so are the English books.).### Response: This problem is asking me to count the number of permutations of two groups of books, where the order within each group matters.
To do this, I can use the multiplication principle: I can first choose how to order the math books, then choose how to order the English books, then choose how to place the two groups on the shelf.
There are 3 math books, so I can order them in 3! = 6 ways.
There are 5 English books, so I can order them in 5! = 120 ways.
Once I have ordered the books within each group, I can treat each group as a single unit.
Then I have two units to place on the shelf, which I can do in 2! = 2 ways.
So by the multiplication principle, the total number of ways to arrange the books is 3! * 5! * 2! = 6 * 120 * 2 = 1440.