There are 3 distinct books on physics, 4 distinct books on chemistry and 5 distinct books on Biology. In how many ways can these books be placed on a shelf if the books on the same subject are to be together?

Options

A. 103680
B. 17280
C. `12!/3!`
D. 60

Answer

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

### Step 1: Treat each subject as a single unit
Since we want the books of the same subject to be together, we can treat each subject as a single unit or block. Therefore, we have three blocks:
- Block 1: Physics (3 books)
- Block 2: Chemistry (4 books)
- Block 3: Biology (5 books)

### Step 2: Calculate the arrangements of the blocks
We can arrange these 3 blocks in different ways. The number of ways to arrange 3 blocks is given by the factorial of the number of blocks:
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 3: Calculate the arrangements within each block
Next, we need to find out how many ways we can arrange the books within each block:
- For Physics, the 3 distinct books can be arranged in:
\[ 3! = 6 \text{ ways} \]
- For Chemistry, the 4 distinct books can be arranged in:
\[ 4! = 24 \text{ ways} \]
- For Biology, the 5 distinct books can be arranged in:
\[ 5! = 120 \text{ ways} \]

### Step 4: Calculate the total arrangements
To find the total arrangements of the books on the shelf, we multiply the number of arrangements of the blocks by the arrangements within each block:
\[ \text{Total arrangements} = (3!) \times (3!) \times (4!) \times (5!) \]
Substituting the values we calculated:
\[ \text{Total arrangements} = 6 \times 6 \times 24 \times 120 \]

### Step 5: Perform the calculations
Now we can calculate:
1. Calculate \(6 \times 6 = 36\)
2. Calculate \(36 \times 24 = 864\)
3. Calculate \(864 \times 120 = 103680\)

Thus, the total number of ways to arrange the books on the shelf is:
\[ \text{Total arrangements} = 103680 \]

### Final Answer:
The total number of ways to arrange the books on the shelf is **103680**.

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