Eleven books, consisting of five Engineering books, four Mathematics books and two Physics books, are arranged in a shelf at random. What is the probability that the books of each kind are all together?

Options

A. `5/1155`
B. `2/1155`
C. `3/1155`
D. `1/1155`

Answer

To solve the problem of finding the probability that all books of each kind are together, we can follow these steps:

### Step 1: Determine the Total Number of Books
We have a total of 11 books:
- 5 Engineering books
- 4 Mathematics books
- 2 Physics books

### Step 2: Calculate the Total Arrangements of the Books
The total number of ways to arrange 11 books is given by the factorial of the total number of books:
\[ 11! = 39916800 \]

### Step 3: Treat Each Subject as a Single Unit
Since we want the books of each kind to be together, we can treat each subject as a single unit. Therefore, we have:
- 1 unit for Engineering (5 books)
- 1 unit for Mathematics (4 books)
- 1 unit for Physics (2 books)

This gives us a total of 3 units to arrange.

### Step 4: Calculate the Arrangements of the Subject Units
The number of ways to arrange these 3 units is:
\[ 3! = 6 \]

### Step 5: Calculate the Arrangements Within Each Subject
Now, we need to consider the arrangements within each subject:
- The 5 Engineering books can be arranged in \(5!\) ways.
- The 4 Mathematics books can be arranged in \(4!\) ways.
- The 2 Physics books can be arranged in \(2!\) ways.

Calculating these:
\[ 5! = 120, \quad 4! = 24, \quad 2! = 2 \]

### Step 6: Calculate the Total Favorable Arrangements
The total number of favorable arrangements where all books of each kind are together is:
\[ 3! \times 5! \times 4! \times 2! = 6 \times 120 \times 24 \times 2 \]

Calculating this step-by-step:
- \(6 \times 120 = 720\)
- \(720 \times 24 = 17280\)
- \(17280 \times 2 = 34560\)

### Step 7: Calculate the Probability
The probability that all books of each kind are together is given by the ratio of favorable arrangements to total arrangements:
\[ P = \frac{\text{Favorable Arrangements}}{\text{Total Arrangements}} = \frac{34560}{39916800} \]

### Step 8: Simplify the Probability
To simplify:
\[ P = \frac{34560}{39916800} = \frac{1}{1155} \]

### Final Answer
Thus, the probability that the books of each kind are all together is:
\[ \frac{1}{1155} \]