n books are to be arranged on a self. These include m volumes of a science book `(m lt n)`. The probability that in any arrangement, the volumes of science books are in ascending order is-

Options

A. `1/(lfloormn)`
B. `1/(lfloorn)`
C. `1/(lfloorm)`
D. `(lfloorm)/(lfloorn)`

Answer

To find the probability that the m volumes of science books are arranged in ascending order when n books are arranged on a shelf, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Total Arrangements**:
- We have a total of n books, which include m volumes of science books and (n - m) other books.
- The total number of ways to arrange n books is given by \( n! \) (n factorial).

2. **Arranging the Science Books**:
- The m volumes of science books can be arranged in any order. The total arrangements of these m books is \( m! \) (m factorial).
- However, we are interested in the specific arrangement where these m books are in ascending order. There is only **1** way to arrange the m books in ascending order.

3. **Calculating the Probability**:
- The probability that the m volumes of science books are in ascending order in any random arrangement of the n books can be calculated as:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{m!} \]
- Here, the number of favorable outcomes is 1 (the ascending order), and the total outcomes is \( m! \) (the total arrangements of the m books).

4. **Final Result**:
- Therefore, the probability that the m volumes of science books are arranged in ascending order is:
\[ \frac{1}{m!} \]