If you join the mid-points of a triangle, you form 4 smaller equilateral triangles as follows

Thus, the shaded triangle (second triangle in the question) has an area exactly $\frac{1}{4}$ of the area of the outer triangle. This process will continue and the next triangle which will be formed from the midpoints of the second triangle will have an area =$\frac{1}{4}$ of the second triangle

Thus this forms an infinite Geometric progression where the common ratio is $\frac{1}{4}$ and the initial term (Area of first triangle)=24

$S=\frac{a}{1-r}=\frac{24}{1-\frac{1}{4}}=\frac{24\times 4}{3}=32$ units. Answer is option (c)