The problem is asking to find the number of ways to arrange the books, given that the books of the same subject are grouped together. You mentioned that there are:

• 5 History books,
• 4 Science books, and
• 3 Math books.

Since the books of the same subject are to be grouped together, we can treat each group of books as a single entity initially. This means we will first arrange the three groups (History, Science, Math).

Step 1: Arrange the groups of books

There are 3 groups, so the number of ways to arrange these 3 groups is: $3! = 3 \times 2 \times 1 = 6$

Step 2: Arrange the books within each group

• The 5 History books can be arranged within their group in $5!$ ways: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

• The 4 Science books can be arranged within their group in $4!$ ways: $4! = 4 \times 3 \times 2 \times 1 = 24$

• The 3 Math books can be arranged within their group in $3!$ ways: $3! = 3 \times 2 \times 1 = 6$

Step 3: Total number of arrangements

The total number of ways to arrange the books, keeping the books of the same subject together, is the product of the arrangements of the groups and the arrangements within each group: $3! \times 5! \times 4! \times 3! = 6 \times 120 \times 24 \times 6$

Let me calculate this for you.The total number of ways to arrange the books, keeping the books of the same subject together, is 103,680.

Would you like a more detailed explanation or further clarification?

Here are 5 related questions that expand on this topic:

1. How would the number of arrangements change if the books of each subject were not grouped together?
2. What if two of the three subjects needed to be grouped, but not the third?
3. How does the factorial operation apply to problems involving permutations and combinations?
4. What if two books of the same subject were identical? How would that affect the calculation?
5. How can this concept be extended to more complex grouping problems in combinatorics?

Tip: Grouping items before arranging them reduces the complexity of the problem, as you treat the groups as single entities first and then handle the arrangements within each group separately.