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 the books on a shelf such that books of the same subject are together, we can follow these steps:

### Step-by-Step Solution:

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

Thus, we have 3 blocks to arrange: [Physics, Chemistry, Biology].

2. **Arrange the Blocks**:
The number of ways to arrange these 3 blocks is given by the factorial of the number of blocks:
\[ \text{Ways to arrange the blocks} = 3! = 6 \]

3. **Arrange the Books Within Each Block**:
- For Physics, the number of ways to arrange the 6 books is \(6!\).
- For Chemistry, the number of ways to arrange the 3 books is \(3!\).
- For Biology, the number of ways to arrange the 4 books is \(4!\).

Therefore, the total arrangements within each block can be calculated as:
\[ \text{Total arrangements within blocks} = 6! \times 3! \times 4! \]

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

5. **Combine the Arrangements**:
Now, we can find the total number of arrangements by multiplying the arrangements of the blocks by the arrangements within each block:
\[ \text{Total arrangements} = 3! \times 6! \times 3! \times 4! = 6 \times 720 \times 6 \times 24 \]

6. **Perform the Final Calculation**:
- First, calculate \(720 \times 6 = 4320\).
- Then, calculate \(4320 \times 24 = 103680\).
- Finally, multiply by 6 (for the arrangement of blocks):
\[ 6 \times 103680 = 622080 \]

### Final Answer:
The total number of ways to arrange the books on the shelf, keeping books of the same subject together, is **622080**.

---