There are 3 works. One is of 3 volumes, one is of 4 volumes and one is of only one and they are placed a random in at shelf. What is the chance that volume of the same work is placed together

Options

A. `1//40`
B. `3//140`
C. `9//70`
D. none

Answer

To solve the problem of finding the probability that volumes of the same work are placed together on a shelf, we can follow these steps:

### Step 1: Understand the Total Number of Volumes
We have three works:
- Work A has 3 volumes (A1, A2, A3)
- Work B has 4 volumes (B1, B2, B3, B4)
- Work C has 1 volume (C1)

Thus, the total number of volumes is:
\[ \text{Total volumes} = 3 + 4 + 1 = 8 \]

### Step 2: Calculate the Total Arrangements
The total number of ways to arrange these 8 volumes is given by the factorial of the total number of volumes:
\[ \text{Total arrangements} = 8! \]

### Step 3: Grouping the Volumes of the Same Work
To find the arrangements where volumes of the same work are together, we can treat each work as a single unit:
- Group A (3 volumes) can be treated as a single unit: [A]
- Group B (4 volumes) can be treated as a single unit: [B]
- Group C (1 volume) can be treated as a single unit: [C]

Now we have 3 groups: [A], [B], [C].

### Step 4: Calculate the Arrangements of Groups
The number of ways to arrange these 3 groups is:
\[ 3! \]

### Step 5: Calculate the Internal Arrangements of Each Group
Within each group, the volumes can be arranged among themselves:
- For Group A (3 volumes): \( 3! \)
- For Group B (4 volumes): \( 4! \)
- For Group C (1 volume): \( 1! \)

Thus, the total arrangements for the groups while keeping the volumes together is:
\[ \text{Favorable arrangements} = 3! \times 3! \times 4! \times 1! \]

### Step 6: Calculate the Probability
Now we can find the probability that volumes of the same work are placed together:
\[ P(\text{same work together}) = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{3! \times 3! \times 4! \times 1!}{8!} \]

### Step 7: Simplifying the Probability
Calculating the factorials:
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 1! = 1 \)
- \( 8! = 40320 \)

Substituting these values into the probability formula:
\[ P(\text{same work together}) = \frac{6 \times 6 \times 24 \times 1}{40320} = \frac{864}{40320} \]

### Step 8: Reducing the Fraction
Now, we simplify \( \frac{864}{40320} \):
\[ \frac{864 \div 864}{40320 \div 864} = \frac{1}{46.875} \approx \frac{1}{47} \]

Thus, the probability that volumes of the same work are placed together is:
\[ P(\text{same work together}) = \frac{3}{140} \]

### Final Answer
The probability that volumes of the same work are placed together is:
\[ \frac{3}{140} \]