Given:

$\mathrm{S}=\{0,1,2,3,\dots ,100\}$

Now, possible ordered pair $\left(x, y\right)\in S$ such that $x\ne y \& x+y=100$ are

$\left(0, 100\right), \left(1, 99\right), \left(2, 98\right), \left(3, 97\right),......, \left(48, 52\right), \left(49, 51\right)$

Therefore, total number of ways of selecting such pairs is $50$.