Example 1 In a certain parliament, there are 30 deputies, and any two deputies are either political enemies or friends. Moreover, each deputy has exactly 6 political enemies. For a committee of 3 deputies, if any two of the three are friends or any two are political enemies, it is called a strange committee. How many strange committees are there? (24th All-Russian Mathematical Olympiad problem)
Solve: This problem has a clear graph theory flavor: either friends or political enemies - connecting edges or not (coloring red or blue); each senator has 6 political enemies - each point leads to 6 red edges; the peculiar committee - monochromatic triangles.

Consider a complete graph with 30 points. Each point represents a senator; if two senators are political enemies, the edge between these two points is colored red; otherwise, it is colored blue. According to the problem, each point exactly leads to 6 red edges. We will calculate the number of monochromatic triangles (subsets) in the graph $(|F|)$. Using the reduction technique, we choose an intermediate variable: calculate the total number of monochromatic angles $S$.

On one hand, for $X$, we can calculate from each point, as each point leads to 6 red edges and 23 blue edges, there are $\mathrm{C}_{6}^{2}+\mathrm{C}_{23}^{2}=268$ monochromatic angles. Therefore, 30 points have $30 \times 268=8040$ monochromatic angles. The monochromatic angles from different points are obviously different, so $S=8040$.

On the other hand, each monochromatic angle is in some triangle. Suppose there are $x$ monochromatic triangles in all triangles, then there are $\mathrm{C}_{30}^{3}-x=4060-x$ triangles with different colors. For each monochromatic triangle, it has 3 monochromatic angles; for each triangle with different colors on all three sides, it has 1 monochromatic angle. Thus, $S=3 x+4060-x=2 x+4060$.

From $8040=S=2 x+4060$, solving for $x$ gives $x=1990$.