A number x is chosen at random from the set S = (1, 2, 3,....,100).Then the probability that the expression `sqrt((x-15)/((x-10)(x-20)))` is a positive real number is

Options

A. `11//25`
B. `7//25`
C. `21//25`
D. `17//25`

Answer

To solve the problem, we need to find the probability that the expression

\[ \sqrt{\frac{x-15}{(x-10)(x-20)}} \]

is a positive real number when \( x \) is chosen from the set \( S = \{1, 2, 3, \ldots, 100\} \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression inside the square root must be greater than 0 for the square root to be a positive real number. Therefore, we need to solve the inequality:

\[ \frac{x-15}{(x-10)(x-20)} > 0 \]

2. **Finding Critical Points**:
The critical points occur when the numerator or denominator is zero:
- Numerator: \( x - 15 = 0 \) → \( x = 15 \)
- Denominator: \( x - 10 = 0 \) → \( x = 10 \)
- Denominator: \( x - 20 = 0 \) → \( x = 20 \)

Thus, the critical points are \( x = 10, 15, 20 \).

3. **Testing Intervals**:
We will test the sign of the expression in the intervals defined by these critical points:
- \( (-\infty, 10) \)
- \( (10, 15) \)
- \( (15, 20) \)
- \( (20, \infty) \)

- For \( x < 10 \) (e.g., \( x = 5 \)):
\[ \frac{5-15}{(5-10)(5-20)} = \frac{-10}{(-5)(-15)} = \frac{-10}{75} < 0 \]
- For \( 10 < x < 15 \) (e.g., \( x = 12 \)):
\[ \frac{12-15}{(12-10)(12-20)} = \frac{-3}{(2)(-8)} = \frac{-3}{-16} > 0 \]
- For \( 15 < x < 20 \) (e.g., \( x = 18 \)):
\[ \frac{18-15}{(18-10)(18-20)} = \frac{3}{(8)(-2)} = \frac{3}{-16} < 0 \]
- For \( x > 20 \) (e.g., \( x = 25 \)):
\[ \frac{25-15}{(25-10)(25-20)} = \frac{10}{(15)(5)} = \frac{10}{75} > 0 \]

4. **Identifying Valid Intervals**:
The expression is positive in the intervals:
- \( (10, 15) \)
- \( (20, \infty) \)

5. **Counting Valid Values**:
- In the interval \( (10, 15) \), the valid integers are \( 11, 12, 13, 14 \) (4 values).
- In the interval \( (20, 100) \), the valid integers are \( 21, 22, \ldots, 100 \). The count is \( 100 - 20 = 80 \) values.

Thus, the total number of favorable outcomes is \( 4 + 80 = 84 \).

6. **Calculating Probability**:
The total number of possible outcomes when choosing \( x \) from \( S \) is 100. Therefore, the probability \( P \) that the expression is a positive real number is:

\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{84}{100} = \frac{21}{25} \]

### Final Answer:
The probability that the expression is a positive real number is

\[ \frac{21}{25}. \]