Scientific Bookkeeping

2! (2 Chemistry books), x 3! (3 Maths books), x 1! (Physics) x 3! (3 subjects) = 72

There are $3$ types of books, so we can arrange the books in $3! = 6$ ways. Since there is only one physics book, it can only be arranged in $1! = 1$ way. The two chemistry books can be arranged in $2! = 2$ ways, and the three math books can be arranged in $3! = 6$ ways.

Then, there are $6 \cdot 1 \cdot 2 \cdot 6 = \boxed{72}$ ways to arrange all of the books.

The number of ways we can arrange the chemistry books is $2 \times 1$ . The number of arrangements for math books is $3 \times 2 \times 1$ . The number of arrangements for physics is just $1$ . Then the number of ways to arrange the subjects is $6 \times 2 \times 1$ . Multiplying all these should should be $\boxed{72}$

The chemistry books can be arranged in 2! ways The math books can be arranged in 3! ways. The physics books can be arranged in 1! ways There are 3! ways in which {Maths, Chemistry, Physics} can be arranged. In each of these 3! arrangements there are 2! . 3! . 1! ways of arranging each of the subject books between themselves.

Thus, required solution = 3! . (2! . 3! . 1!)

There are six ways Marina can order the groups: CMP CPM MCP ... PMC

We can calculate this by finding 3P3, or 3!

Within these groups, using the same system, she has 2 ways of organizing her chemistry books, and 6 ways of organizing her math books. Thus, there are 12 ways of organizing her books in these "sub-orderings"

Multiplying the number of higher level orderings with the number of lower level orderings, and we get 72.

The number of way to arrange only the chemistry books is $2! = 2$

The number of way to arrange only the maths books is $3! = 6$

The number of way to arrange only the physics book is $1! = 1$

Consider the chemistry, maths and physics books are of one set(since they are need to be next to each other) respectively, so there are 3 sets of books and the number of way to arrange them is $3! = 6$

Hence, the number of way to arrange the books for each class to be next to each other = $2 \times 6 \times 1 \times 6 =72$

If the books for each class must be next to each other,then we can consider them as 1 book (for now).So there are $3 \times 2 =6$ arrangements.There are 2 arrangements for the chemistry books and 6 for the math books.So the total number of arrangements is $6 \times 6 \times 2 = 72$

Let us group the the six books into three groups, namely chemistry, math, and physics. There are 3! ways to arrange these three groups. Furthermore, we have 2 chemistry books and 3 math books. The chemistry books may be arranged in 2! ways, while the math books may be arranged in 3! ways. Thus, the total number of arrangements is (3!)(2!)(3!)=72.

First, let me write down the final equation to find the number of permutations:

(3p3) $\times$ (2p2) $\times$ (3p3) $\times$ (1p1) , (where p represents permutation ).

3p3 - We must keep subjects together. Since there are 3 subjects, there are 3p3 ways to arrange the categories of books.

2p2 - There are 2 books for chemistry , thus 2p2 ways to arrange these books within the chemistry group of books

Similarly, there are 3p3 and 1p1 ways to arrange the math and physics book(s) respectively within its own group.

Therefore, the total number of permutations is:

(3p3) $\times$ (2p2) $\times$ (3p3) $\times$ (1p1) = 72

We first consider the number of ways to arrange the books as sets, grouped according to class. There are 3 sets, so this gives us 3! = 6. Then we count the number of ways to arrange the books in each set: the chemistry books, 2! = 2, the math books, 3! = 6, and the physics book, 1! = 1. So the total number of ways is 6 * 2 * 6 * 1 = 72.

To arrange 2 chemistry books, we have 2! ways.

To arrange 3 mathematics books, we have 3! ways.

To arrange 1 physics book, we have 1! way.

Then, treat chemistry books, mathematics books and physics book as 3 group to be arranged since the books for each class need to be next to each other. So there is 3! ways.

Multiply all possible ways together 2!x3!x1!x3!=72

So, total ways = 72

The number of ways to arrange the books in Chemistry is 2!, in Math is 3!, and in Physics is 1!. The number of ways to arrange all of the 3 subjects is 3!. Therefore, the number of ways to arrange the books in order for each class to be next to each other is 3! x (3! x 2! x 1!) = 6 x (6 x 2) = 6 x 12 = 72

There are 3 subjects, that is. Chemistry, Math and Physics. The 2 Chemistry books is taken as 1 subject, 3 Math books as 1 subjects, and 1 Physics book, That is Permutation of 3 taken 3 = 6 Since the 2 CHEMISTRY books can be arranged next to each other, that is Permutation of 2 taken 2 = 2. And the 3 MATH books can be arranged next to each other, That is Permutation of 3 taken 3 = 6

According to the Product Rule, we have:

6 ways X 2 ways X 6 ways = 72 ways .

Taking the physics, the chemistry and math as a single box, we have 3! arrangements. And the chemistry books can be arranged in 2! ways, the math books in 3! ways and the physics book in 1! way. Hence, total number of arrangements is 3! 3! 2!*1!=72.