`x_(1) , x_(2) , x_(3)`… . `x_(50)` are fifty real numbers such that `x_(r) lt x_(r + 1)` for r = 1 ,2 , 3 …….. , 49 . Five numbers out of these are picked up at random . The probability that the five numbers have `x_(20)` as the middle number is :

Options

A. `(""^(20) C_(2) xx ""^(30) C_(2))/(""^(20) C_(5))`
B. `(""^(30)C_(2) xx ""^(19) C_(2))/(""^(50) C_(5))`
C. `(""^(19) C_(2) xx ""^(30) C_(3))/(""^(50) C_(5))`
D. None of these

Answer

To solve the problem, we need to find the probability that when we randomly select five numbers from a set of 50 ordered real numbers, \(x_1, x_2, x_3, \ldots, x_{50}\), the middle number is \(x_{20}\).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 50 real numbers arranged in increasing order. We need to select 5 numbers such that \(x_{20}\) is the middle number. The middle number in a selection of 5 numbers is the 3rd number when arranged in increasing order.

2. **Total Ways to Select 5 Numbers**:
The total number of ways to select 5 numbers from 50 is given by the combination formula:
\[ \text{Total ways} = \binom{50}{5} \]

3. **Condition for \(x_{20}\) to be the Middle Number**:
For \(x_{20}\) to be the middle number, we need to select:
- 2 numbers from the first 19 numbers (\(x_1, x_2, \ldots, x_{19}\)) which are less than \(x_{20}\).
- 2 numbers from the numbers after \(x_{20}\) (i.e., \(x_{21}, x_{22}, \ldots, x_{50}\)) which are greater than \(x_{20}\).

4. **Calculating the Number of Ways to Select 2 Numbers**:
- The number of ways to choose 2 numbers from the first 19 numbers:
\[ \text{Ways from } x_1 \text{ to } x_{19} = \binom{19}{2} \]
- The number of ways to choose 2 numbers from the 30 numbers after \(x_{20}\) (i.e., \(x_{21} \text{ to } x_{50}\)):
\[ \text{Ways from } x_{21} \text{ to } x_{50} = \binom{30}{2} \]

5. **Total Ways for \(x_{20}\) to be the Middle Number**:
The total number of favorable ways to select the 5 numbers such that \(x_{20}\) is the middle number is:
\[ \text{Favorable ways} = \binom{19}{2} \times \binom{30}{2} \]

6. **Calculating the Probability**:
The probability \(P\) that \(x_{20}\) is the middle number is given by the ratio of the favorable outcomes to the total outcomes:
\[ P = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{\binom{19}{2} \times \binom{30}{2}}{\binom{50}{5}} \]

### Final Expression:
Thus, the probability that the five numbers have \(x_{20}\) as the middle number is:
\[ P = \frac{\binom{19}{2} \times \binom{30}{2}}{\binom{50}{5}} \]