There are three works,one consisting of 3 volumes, one of 4, and the other of 1 volume. They are placed on a shelf at random. Find the chance that volumes of the same works are all together.

Answer

To solve the problem step by step, we need to find the probability that the volumes of the same works are all together when placed on a shelf.

### Step 1: Identify the total number of volumes
We have three works:
- Work 1: 3 volumes
- Work 2: 4 volumes
- Work 3: 1 volume

Total volumes = 3 + 4 + 1 = 8 volumes.

### Step 2: Calculate the total arrangements of the volumes
The total arrangements of 8 volumes can be calculated using the factorial of the total number of volumes:
\[ \text{Total arrangements} = 8! = 40320 \]

### Step 3: Group the volumes of the same work together
To find the arrangements where volumes of the same work are together, we can treat each work as a single unit or block:
- Block for Work 1 (3 volumes)
- Block for Work 2 (4 volumes)
- Block for Work 3 (1 volume)

Now we have 3 blocks to arrange:
- Total blocks = 3!

### Step 4: Calculate the arrangements within each block
Now we need to arrange the volumes within each block:
- Work 1 (3 volumes): can be arranged in \(3!\) ways.
- Work 2 (4 volumes): can be arranged in \(4!\) ways.
- Work 3 (1 volume): can be arranged in \(1!\) way.

### Step 5: Calculate the total arrangements with blocks
The total arrangements where the volumes of the same work are together can be calculated as:
\[ \text{Arrangements with blocks} = 3! \times 3! \times 4! \times 1! \]
Calculating this:
\[ 3! = 6, \quad 4! = 24, \quad 1! = 1 \]
So,
\[ \text{Arrangements with blocks} = 3! \times 3! \times 4! \times 1! = 6 \times 24 \times 1 = 144 \]

### Step 6: Calculate the probability
Now, we can find the probability that the volumes of the same works are all together:
\[ \text{Probability} = \frac{\text{Arrangements with blocks}}{\text{Total arrangements}} = \frac{144}{40320} \]

### Step 7: Simplify the probability
To simplify:
\[ \frac{144}{40320} = \frac{1}{280} \]

### Final Answer
The probability that the volumes of the same works are all together is \(\frac{1}{280}\).