How to solve this confusing permutation problem related to arrangement of books?

Question

Ms. Jones has 10 books that she is going to put on her bookshelf. Of these, 4 are mathematics books, 3 are chemistry books, 2 are history books, and 1 is a language book. Ms. Jones wants to arrange her books so that all the books dealing with the same subject are together on the shelf. How many different arrangements are possible?

I am completely lost. I know there are 4! 3! 2! 1! arrangements such that the mathematics books are first in line, then the chemistry books, then the history books, and then the language book.

But this is not the answer.

Answer 1

you have already excellent answers so i will try to give you a nice way on how to see the problem.

Suppose that the books of the same category are all in a custody,so we have a custody for math,chemistry,history and language: $4$ custodies in total.

Now the question is: in how many ways can you arrange those 4 custodies? The answer is clearly $4!$. Now what is left is the number of arrangements of books in the same custody,therefore we have $4!\cdot 3! \cdot 2! \cdot 1!$ and you can do that for every arrangement of the $4$ custodies .,therefore you have finally $4!(4! \cdot 3! \cdot 2! \cdot 1!) $

Answer 2

You are almost there, but you need to multiply the arrangement for the categories. There are $4$ categories so the final answer should be $4!(4!3!2!1!)$.

Answer 3

You treat books of same subject as one object (since they have to be together) so you have $ 4!$ options. And then comes your part in which you permute books of same subject. So the final result is $4!4!3!2!1!$.

Answer 4

HINT:

$$\underbrace{\underbrace{M_1M_2M_3M_4}_{4!}\mid\underbrace{C_1C_2C_3}_{3!}\mid\underbrace{H_1H_2}_{2!}\mid\underbrace{L_1}_{1!}}_{4!}$$

Answer 5

you use permutations to solve this because we are talking about different arrangements here. since there are 4 mathematics books you can arrange those 4 using = 4! 3 chemistry books you can arrange that bundle using = 3! 2 history books can be arranged using = 2! 1 language book can be arranged in only one way so it's = 1! you can treat the all the math books as one bundle and if you consider the other bundles all together you have 4 bundles. so there are 4! ways of arranging those bundles.

total arrangement =[ 4! 3! 2! 1! ] 4! = 6912