Given
![m+n](/media/m/1/2/c/12cfeebf074af065b2efa21bce4eb0fe.png)
numbers:
![b_j](/media/m/e/3/2/e322179d9a2a88fdb932657fa68661a3.png)
,
![j = 1,2, \ldots, n,](/media/m/6/4/3/6432544bc1176ba2067477dd276689b7.png)
determine the number of pairs
![(a_i,b_j)](/media/m/2/5/7/25740b4c9abe19df10640e45b190ee21.png)
for which
![|i-j| \geq k,](/media/m/c/2/4/c2416baf0831d33d5f48e58634207dce.png)
where
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
is a non-negative integer.
%V0
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.