To solve the problem, we need to find the probability that the absolute difference between two distinct integers \( x \) and \( y \), chosen from the set of integers from 0 to 10, is less than or equal to 5.
### Step-by-Step Solution:
1. **Identify the Set of Integers**:
The integers \( x \) and \( y \) can take values from the set \( \{0, 1, 2, \ldots, 10\} \). This set contains 11 integers.
2. **Calculate Total Outcomes**:
Since \( x \) and \( y \) are chosen without replacement, the total number of ways to choose \( x \) and \( y \) is given by:
\[
\text{Total Outcomes} = 11 \times 10 = 110
\]
(We have 11 choices for \( x \) and 10 remaining choices for \( y \)).
3. **Define the Condition**:
We need to find the probability that \( |x - y| \leq 5 \). This can be rewritten as:
\[
-5 \leq x - y \leq 5
\]
This means that \( x \) can be at most 5 units away from \( y \) in either direction.
4. **Count Favorable Outcomes**:
We will count the pairs \( (x, y) \) that satisfy \( |x - y| \leq 5 \):
- For each possible value of \( y \) from 0 to 10, we will determine the valid range for \( x \):
- If \( y = 0 \): \( x \) can be \( 0, 1, 2, 3, 4, 5 \) (6 options)
- If \( y = 1 \): \( x \) can be \( 0, 1, 2, 3, 4, 5, 6 \) (7 options)
- If \( y = 2 \): \( x \) can be \( 0, 1, 2, 3, 4, 5, 6, 7 \) (8 options)
- If \( y = 3 \): \( x \) can be \( 0, 1, 2, 3, 4, 5, 6, 7, 8 \) (9 options)
- If \( y = 4 \): \( x \) can be \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \) (10 options)
- If \( y = 5 \): \( x \) can be \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \) (11 options)
- If \( y = 6 \): \( x \) can be \( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \) (10 options)
- If \( y = 7 \): \( x \) can be \( 2, 3, 4, 5, 6, 7, 8, 9, 10 \) (9 options)
- If \( y = 8 \): \( x \) can be \( 3, 4, 5, 6, 7, 8, 9, 10 \) (8 options)
- If \( y = 9 \): \( x \) can be \( 4, 5, 6, 7, 8, 9, 10 \) (7 options)
- If \( y = 10 \): \( x \) can be \( 5, 6, 7, 8, 9, 10 \) (6 options)
Now, we sum these options:
\[
6 + 7 + 8 + 9 + 10 + 11 + 10 + 9 + 8 + 7 + 6 = 91
\]
5. **Calculate the Probability**:
The probability \( P \) that \( |x - y| \leq 5 \) is given by the ratio of favorable outcomes to total outcomes:
\[
P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{91}{110}
\]
### Final Answer:
The probability that \( |x - y| \leq 5 \) is \( \frac{91}{110} \).
