There is 4 volume encyclopaedia among 40 books arranged on a shelf in a random order. If the volumes are not necessarily kept side by side, the probability that they occur in increasing order from left to right is :

Options

A. `1/24`
B. `1/12`
C. `1/10`
D. none of these

Answer

To solve the problem of finding the probability that the four volumes of an encyclopedia occur in increasing order from left to right among 40 books, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Total Books**: We have a total of 40 books, which include 4 volumes of an encyclopedia and 36 other books.

2. **Total Arrangements of Books**: The total number of ways to arrange 40 books is given by \(40!\) (40 factorial).

3. **Choosing Positions for Encyclopedia Volumes**: We need to choose 4 positions from the 40 available positions for the encyclopedia volumes. This can be calculated using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Thus, we have:
\[ \text{Ways to choose positions} = \binom{40}{4} \]

4. **Arranging the Other Books**: The remaining 36 books can be arranged in any order. The number of arrangements for these 36 books is \(36!\).

5. **Favorable Arrangements for Encyclopedia Volumes**: The 4 encyclopedia volumes can only be arranged in one specific order (increasing order). Therefore, there is only 1 way to arrange these 4 volumes once their positions are chosen.

6. **Calculating Total Favorable Outcomes**: The total number of favorable outcomes (where the encyclopedia volumes are in increasing order) is:
\[ \text{Favorable Outcomes} = \binom{40}{4} \times 36! \]

7. **Calculating the Probability**: The probability \( P \) that the encyclopedia volumes occur in increasing order is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{\binom{40}{4} \times 36!}{40!} \]

8. **Simplifying the Probability**: We can simplify the expression:
\[ P = \frac{\binom{40}{4} \times 36!}{40!} = \frac{\frac{40!}{4! \times 36!} \times 36!}{40!} = \frac{1}{4!} = \frac{1}{24} \]

### Final Answer:
Thus, the probability that the four volumes occur in increasing order from left to right is:
\[ \frac{1}{24} \]