There are 6 books of physics , 3 of chemistry and 4 of biology . Number of ways in which these bokks be placed on a shelf if the books of the ame subject are to be together is

Options

A. 622080
B. 888000
C. 222000
D. 413080

Answer

To solve the problem of arranging books on a shelf with the condition that books of the same subject must be together, we can follow these steps:

### Step-by-Step Solution:

1. **Group the Books by Subject**:
- We have 6 Physics books, 3 Chemistry books, and 4 Biology books.
- We can treat each subject's books as a single unit or "block". Thus, we have three blocks: Physics, Chemistry, and Biology.

2. **Arrange the Subject Blocks**:
- The number of ways to arrange these 3 blocks (Physics, Chemistry, Biology) is given by the factorial of the number of blocks.
- This can be calculated as \(3!\).

3. **Arrange the Books Within Each Block**:
- For the Physics block, the 6 books can be arranged in \(6!\) ways.
- For the Chemistry block, the 3 books can be arranged in \(3!\) ways.
- For the Biology block, the 4 books can be arranged in \(4!\) ways.

4. **Calculate the Total Arrangements**:
- The total number of arrangements is the product of the arrangements of the blocks and the arrangements within each block.
- This can be expressed as:
\[ \text{Total Arrangements} = 3! \times 6! \times 3! \times 4! \]

5. **Calculate the Factorials**:
- Now we calculate the factorials:
- \(3! = 6\)
- \(6! = 720\)
- \(4! = 24\)

6. **Substitute the Values**:
- Substitute these values into the equation:
\[ \text{Total Arrangements} = 6 \times 720 \times 6 \times 24 \]

7. **Perform the Multiplication**:
- First, calculate \(6 \times 720 = 4320\).
- Next, calculate \(4320 \times 6 = 25920\).
- Finally, calculate \(25920 \times 24 = 622080\).

8. **Final Answer**:
- Therefore, the total number of ways to arrange the books on the shelf, with the condition that books of the same subject are together, is \(622080\).