Final answer:
There are 622080 ways to arrange the books on a shelf with the constraint that books of the same subject must be together. Calculate this by multiplying the factorial of the number of units (3!) with the factorial of the number of books in each subject (6!, 3!, and 4!).
Step-by-step explanation:
To solve the problem of arranging books on a shelf where books of the same subject must be together, we treat each set of same-subject books as a single unit.
This means we have three units to arrange: physics books, chemistry books, and biology books.
There are 3! or 6 ways to arrange these three units on a shelf.
Within each unit, we can arrange the books in any order.
For the physics books, there are 6! or 720 ways to arrange them.
For the chemistry books, there are 3! or 6 ways to arrange them, and for the biology books, there are 4! or 24 ways to arrange them.
The total number of arrangements is the product of these individual arrangements:
Total arrangements = (Number of ways to arrange the units) × (Number of arrangements for physics books) × (Number of arrangements for chemistry books) × (Number of arrangements for biology books)
Total arrangements = 3! × 6! × 3! × 4!
= 6 × 720 × 6 × 24
Calculating this gives us 6 × 720 × 6 × 24 = 622080, which corresponds to option A.
Therefore, 622080 is the number of ways in which these books can be placed on a shelf with the constraint that books of the same subject are to be together.