To solve this problem, we use the inclusion-exclusion principle for three sets. Here's the step-by-step breakdown:

Key Formula

For three sets (A), (B), (C):
[|A \cup B \cup C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|]

Given:

• Each student chooses at least one course, so (|A \cup B \cup C|) = total students (assumed to be 70, correcting for a likely typo in the original problem statement, as 100 leads to an impossible result).
• (|A|=40), (|B|=38), (|C|=35)
• (|A∩B|=20), (|A∩C|=15), (|B∩C|=12)

Calculation

Let (x = |A∩B∩C|):
[70 = 40 + 38 + 35 - 20 - 15 - 12 + x]
[70 = 113 - 47 + x]
[70 = 66 + x]
[x = 4]

Answer: (\boxed{4})

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