To solve the problem of finding the number of ordered pairs \((x, y)\) such that \(x, y \in \{0, 1, 2, \ldots, 10\}\) and \(|x - y| > 5\), we can break it down into two cases: when \(x > y\) and when \(x < y\).
### Step-by-Step Solution:
**Step 1: Analyze the condition \(|x - y| > 5\)**
This condition can be split into two inequalities:
1. \(x - y > 5\) (which implies \(x > y + 5\))
2. \(y - x > 5\) (which implies \(y > x + 5\))
**Step 2: Case 1 - \(x > y\)**
For this case, we need to satisfy \(x > y + 5\).
- The minimum value for \(y\) is \(0\). If \(y = 0\), then \(x > 5\). Thus, the possible values for \(x\) are \(6, 7, 8, 9, 10\).
- If \(y = 1\), then \(x > 6\). The possible values for \(x\) are \(7, 8, 9, 10\).
- If \(y = 2\), then \(x > 7\). The possible values for \(x\) are \(8, 9, 10\).
- If \(y = 3\), then \(x > 8\). The possible values for \(x\) are \(9, 10\).
- If \(y = 4\), then \(x > 9\). The only possible value for \(x\) is \(10\).
- If \(y = 5\), \(x\) must be greater than \(10\), which is not possible.
Now, let's count the pairs:
- For \(y = 0\): 5 options for \(x\) (6, 7, 8, 9, 10)
- For \(y = 1\): 4 options for \(x\) (7, 8, 9, 10)
- For \(y = 2\): 3 options for \(x\) (8, 9, 10)
- For \(y = 3\): 2 options for \(x\) (9, 10)
- For \(y = 4\): 1 option for \(x\) (10)
- For \(y = 5\): 0 options for \(x\)
Total pairs when \(x > y\):
\[ 5 + 4 + 3 + 2 + 1 + 0 = 15 \]
**Step 3: Case 2 - \(x < y\)**
For this case, we need to satisfy \(y > x + 5\).
- The minimum value for \(x\) is \(0\). If \(x = 0\), then \(y > 5\). Thus, the possible values for \(y\) are \(6, 7, 8, 9, 10\).
- If \(x = 1\), then \(y > 6\). The possible values for \(y\) are \(7, 8, 9, 10\).
- If \(x = 2\), then \(y > 7\). The possible values for \(y\) are \(8, 9, 10\).
- If \(x = 3\), then \(y > 8\). The possible values for \(y\) are \(9, 10\).
- If \(x = 4\), then \(y > 9\). The only possible value for \(y\) is \(10\).
- If \(x = 5\), \(y\) must be greater than \(10\), which is not possible.
Now, let's count the pairs:
- For \(x = 0\): 5 options for \(y\) (6, 7, 8, 9, 10)
- For \(x = 1\): 4 options for \(y\) (7, 8, 9, 10)
- For \(x = 2\): 3 options for \(y\) (8, 9, 10)
- For \(x = 3\): 2 options for \(y\) (9, 10)
- For \(x = 4\): 1 option for \(y\) (10)
- For \(x = 5\): 0 options for \(y\)
Total pairs when \(x < y\):
\[ 5 + 4 + 3 + 2 + 1 + 0 = 15 \]
**Step 4: Combine the results**
The total number of ordered pairs \((x, y)\) such that \(|x - y| > 5\) is:
\[ 15 + 15 = 30 \]
### Final Answer:
The total number of ordered pairs \((x, y)\) such that \(|x - y| > 5\) is **30**.
