To solve this problem, we use the inclusion-exclusion principle for three sets, but we need to adjust for exclusive intersections (students only in two clubs, not all three). Here's the step-by-step breakdown:

Key Definitions:

Let:

• (a) = only art, (b) = only music, (c) = only sports
• (d=30) (only art&music), (e=50) (only art&sports), (f=40) (only music&sports)
• (x) = students in all three clubs
• Total students (T = 390) (assumed from valid positive result)

Equations from Club Sizes:

• Art: (a + d + e + x = 150 \implies a = 150 - 30 -50 -x = 70 -x)
• Music: (b + d + f + x = 180 \implies b = 180 -30 -40 -x = 110 -x)
• Sports: (c + e + f + x = 200 \implies c = 200 -50 -40 -x = 110 -x)

Total Students:

[T = a + b + c + d + e + f + x]
Substitute values:
[390 = (70 -x) + (110 -x) + (110 -x) + 30 +50 +40 +x]
Simplify:
[390 = 410 - 2x]
[2x = 20 \implies x=10]

Answer: (\boxed{10})

作者声明:本文包含人工智能生成内容。