Two integers x and y are chosen with replacement out of the set {0, 1, 2, 3, ... 10). Then the probability that |x - y । > 5 is:

Options

A. `7/11`
B. `40/121`
C. `35/121`
D. `30/121`

Answer

To solve the problem of finding the probability that |x - y| > 5 when x and y are chosen from the set {0, 1, 2, 3, ..., 10}, we can follow these steps:

### Step 1: Identify the total number of outcomes
Since x and y are chosen from the set {0, 1, 2, 3, ..., 10}, there are 11 possible integers (0 through 10). Since the selection is with replacement, the total number of possible pairs (x, y) is:
\[ \text{Total Outcomes} = 11 \times 11 = 121 \]

**Hint:** Remember that when choosing with replacement, the same number can be chosen more than once.

### Step 2: Determine the condition |x - y| > 5
The condition |x - y| > 5 means that the difference between x and y must be greater than 5. This can happen in two scenarios:
1. \( x - y > 5 \)
2. \( y - x > 5 \)

This translates to:
1. \( x > y + 5 \)
2. \( y > x + 5 \)

### Step 3: Find the valid pairs (x, y)
Now we will find the pairs (x, y) that satisfy these conditions.

**For \( x > y + 5 \):**
- If \( y = 0 \), then \( x \) can be 6, 7, 8, 9, 10 (5 options)
- If \( y = 1 \), then \( x \) can be 7, 8, 9, 10 (4 options)
- If \( y = 2 \), then \( x \) can be 8, 9, 10 (3 options)
- If \( y = 3 \), then \( x \) can be 9, 10 (2 options)
- If \( y = 4 \), then \( x \) can be 10 (1 option)
- If \( y \geq 5 \), there are no valid x values since x can only go up to 10.

Calculating the total for this case:
\[ 5 + 4 + 3 + 2 + 1 = 15 \]

**For \( y > x + 5 \):**
- If \( x = 0 \), then \( y \) can be 6, 7, 8, 9, 10 (5 options)
- If \( x = 1 \), then \( y \) can be 7, 8, 9, 10 (4 options)
- If \( x = 2 \), then \( y \) can be 8, 9, 10 (3 options)
- If \( x = 3 \), then \( y \) can be 9, 10 (2 options)
- If \( x = 4 \), then \( y \) can be 10 (1 option)
- If \( x \geq 5 \), there are no valid y values since y can only go up to 10.

Calculating the total for this case:
\[ 5 + 4 + 3 + 2 + 1 = 15 \]

### Step 4: Calculate the total favorable outcomes
Adding both cases together gives us the total number of favorable outcomes:
\[ \text{Favorable Outcomes} = 15 + 15 = 30 \]

### Step 5: Calculate the probability
Now we can calculate the probability:
\[ P(|x - y| > 5) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{30}{121} \]

### Final Answer
The probability that |x - y| > 5 is:
\[ \frac{30}{121} \]

---