How many ways can you arrange 5 mathematics books 4 science books and 3 English books on a shelf such that books of the same subject are kept together?

Six math books, five science books, and two English books are to be arranged on a shelf, so that books of the same subject must be together. How many ways can this be done? * ================ Math books: 6! ways to arrange them Science books: 5! ways to arrange them English books: 2! ways to arrange them 6!*5!*2! = 720*120*2 = 172800 ways to arrange the Math, Science, and English books together, in that particular subject order. You can also arrange the three subjects in 3! ways, bringing the total number of arrangements to:

(172800)*3! = 172800*6 =

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