1. Form of the triangle

   Since one vertex is at the origin and the other two vertices lie on the coordinate axes, the triangle must have vertices of the form $O(0,0),\quad A(a,0),\quad B(0,b)$ where $a,b$ are nonzero integers.

   Because the points can lie on either positive or negative sides of the axes, $a$ and $b$ may be positive or negative, but not zero.

2. Area condition

   The area of triangle $OAB$ is $\text{Area} = \frac{1}{2}|ab|$

   Given that the area is $50$, we get $\frac{1}{2}|ab| = 50$ $|ab| = 100$

3. Count integral possibilities for $|a|,|b|$

   So we need the number of integer pairs $(a,b)$ such that $|a||b| = 100$

   Let $x=|a|,\quad y=|b|$ Then $xy=100$ with $x,y$ positive integers.

   Factorizing: $100 = 2^2\cdot 5^2$ Therefore the number of positive divisors of $100$ is $d(100)=(2+1)(2+1)=9$

   Hence there are $9$ ordered positive pairs $(x,y)$ satisfying $xy=100$.

   These are: $(1,100),(2,50),(4,25),(5,20),(10,10),(20,5),(25,4),(50,2),(100,1)$

4. Include sign choices

   For each positive pair $(x,y)$, we can choose $a=\pm x,\qquad b=\pm y$ independently.

   So each pair gives $4$ triangles: $(x,y),\ (x,-y),\ (-x,y),\ (-x,-y)$

   Therefore total number of triangles is $9\times 4 = 36$

5. Check options

   • A: $9$ — counts only positive divisor pairs, ignores signs.
   • B: $18$ — incomplete sign counting.
   • C: $36$ — correct.
   • D: $32$ — incorrect.

Hence, the number of elements in $S$ is $\boxed{36}$