Four digit number is formed from all possible ways. The probability that a number xyzt chosen from the number satisfy `x lt y = z gt t` is (A) `2/75` (B) `3/75` (C) ` 2/25` (D) `3/25`

Options

A. `2/75`
B. `3/75`
C. `2/25`
D. `3/25`

Answer

To solve the problem, we need to determine the probability that a randomly chosen four-digit number \( xyzt \) satisfies the condition \( x < y = z > t \).

### Step-by-Step Solution:

1. **Determine the total number of four-digit numbers**:
- The first digit \( x \) cannot be 0 (it must be from 1 to 9), so there are 9 options for \( x \).
- The digits \( y, z, t \) can be any digit from 0 to 9, giving us 10 options for each of these digits.
- Therefore, the total number of four-digit numbers is:
\[ \text{Total numbers} = 9 \times 10 \times 10 \times 10 = 9000 \]

**Hint**: Remember that the first digit of a four-digit number cannot be zero.

2. **Identify the conditions**:
- We need to satisfy the conditions \( x < y \) and \( y = z \) and \( z > t \).
- This implies \( x < y \) and \( y > t \).

3. **Count the favorable cases**:
- For any fixed value of \( y \), \( x \) can take values from 1 to \( y-1 \) (i.e., \( y-1 \) options).
- Since \( z = y \), we need \( t < y \) (i.e., \( y \) options for \( t \)).
- Therefore, for each \( y \), the number of valid combinations of \( (x, y, z, t) \) is:
\[ \text{Favorable cases for fixed } y = (y-1) \times y \]

4. **Calculate the total favorable cases**:
- We need to sum this for all possible values of \( y \) from 2 to 9 (since \( y \) must be at least 2 for \( x < y \)):
\[ \text{Total favorable cases} = \sum_{y=2}^{9} (y-1) \times y \]
- Calculating each term:
- For \( y = 2 \): \( 1 \times 2 = 2 \)
- For \( y = 3 \): \( 2 \times 3 = 6 \)
- For \( y = 4 \): \( 3 \times 4 = 12 \)
- For \( y = 5 \): \( 4 \times 5 = 20 \)
- For \( y = 6 \): \( 5 \times 6 = 30 \)
- For \( y = 7 \): \( 6 \times 7 = 42 \)
- For \( y = 8 \): \( 7 \times 8 = 56 \)
- For \( y = 9 \): \( 8 \times 9 = 72 \)
- Summing these values:
\[ 2 + 6 + 12 + 20 + 30 + 42 + 56 + 72 = 240 \]

5. **Calculate the probability**:
- The probability \( P \) that a randomly chosen four-digit number satisfies the condition is given by:
\[ P = \frac{\text{Total favorable cases}}{\text{Total cases}} = \frac{240}{9000} \]
- Simplifying this fraction:
\[ P = \frac{240 \div 240}{9000 \div 240} = \frac{1}{37.5} = \frac{2}{75} \]

### Final Answer:
The probability that a number \( xyzt \) satisfies \( x < y = z > t \) is \( \frac{2}{75} \).