Inclusion-Exclusion (3 sets)

When you add |A|, |B|, and |C|, any element that lies in more than one set gets counted more than once. Subtracting the pairwise intersections removes those extra counts, but elements in all three sets were subtracted too many times, so |A\cap B\cap C| must be added back once to restore the correct total. This is the three-set version of inclusion-exclusion: count everything once, correct the overcounting, then fix the overcorrection.

Use this when you need the size of a combined group made from three overlapping categories, such as students taking multiple classes, survey respondents with overlapping traits, or items satisfying several conditions. It is especially useful in probability and counting problems where direct counting would double count overlaps.

• Forgetting to add back the triple intersection after subtracting pairwise overlaps
• Using the formula for probabilities without converting counts to probabilities or vice versa
• Treating the pairwise intersections as disjoint and subtracting them in the wrong way