Question: "In how many ways can 2 different history books, 5 different math books, and 4 different novels be arranged on a shelf if the books of each type must be together?"
In this question, sequence of the books is not important, therefore:
- For the 2 history books: 2 ways to arrange them (AB and BA), or $2!$
- For the 5 math books: $5*4*3*2*1 = 5!$ ways to arrange them, or 120
- For the 4 novels: $4*3*2*1 = $4!$ ways to arrange them, or 24
Think like this:
- For the history books (assuming we only look at the history books): 2 options for the first slot, and 1 for the last
- For the math books (again, only look at the math books): 5 options for the first slot, $5-1=4$ for the second slot, $5-2=3$ for the third and so on
- The same for the novels
We also have three types of books, so, the order of first-to-appear is, by the same logic, 3!
Therefore, in the the end you have $2!*5!*4!*3!=34560$ ways to arrange those books