Given $m+n$ numbers: $a_i,$ $i = 1,2, \ldots, m,$ $b_j$, $j = 1,2, \ldots, n,$ determine the number of pairs $(a_i,b_j)$ for which $|i-j| \geq k,$ where $k$ is a non-negative integer.
To determine the number of pairs \((a_i, b_j)\) for which \(|i-j| \geq k\), we need to consider two cases: \(k \geq 1\) and \(k = 0\).

### Case 1: \(k \geq 1\)

1. **Define the sets \(A_i\) and \(B_i\):**
   - For each \(i\) from 1 to \(m\), define \(A_i\) as the number of valid \(b_j\) such that \(|i-j| \geq k\).
   - For each \(j\) from 1 to \(n\), define \(B_i\) as the number of valid \(a_i\) such that \(|i-j| \geq k\).

2. **Calculate \(A_i\):**
   - For a fixed \(i\), \(b_j\) is valid if \(j \leq i - k\) or \(j \geq i + k\).
   - The number of such \(j\) values is:
     \[
     A_i = \begin{cases} 
     n - (2k - 1) & \text{if } 1 \leq i \leq k \\
     n - k & \text{if } k < i \leq n - k \\
     n - (2k - 1) & \text{if } n - k < i \leq n \\
     0 & \text{otherwise}
     \end{cases}
     \]

3. **Calculate \(B_i\):**
   - For a fixed \(j\), \(a_i\) is valid if \(i \leq j - k\) or \(i \geq j + k\).
   - The number of such \(i\) values is:
     \[
     B_i = \begin{cases} 
     m - (2k - 1) & \text{if } 1 \leq j \leq k \\
     m - k & \text{if } k < j \leq m - k \\
     m - (2k - 1) & \text{if } m - k < j \leq m \\
     0 & \text{otherwise}
     \end{cases}
     \]

4. **Sum the valid pairs:**
   - The total number of valid pairs is the sum of all \(A_i\) and \(B_i\):
     \[
     \sum_{i=1}^{m} A_i + \sum_{i=1}^{n} B_i
     \]

### Case 2: \(k = 0\)

1. **All pairs are valid:**
   - When \(k = 0\), the condition \(|i-j| \geq k\) is always true.
   - Therefore, the number of valid pairs is simply the product of the number of \(a_i\) and \(b_j\):
     \[
     mn
     \]

The final answer is \( \boxed{ \sum_{i=1}^{m} A_i + \sum_{i=1}^{n} B_i } \) for \(k \geq 1\) and \(mn\) for \(k = 0\).