2. (Average) For the upcoming semester, 100 math majors can take up to two out of five math electives. Suppose 22 will not take any math elective in the coming semester. Also,
- 7 will take Algebraic Number Theory and Galois Theory
- 12 will take Galois Theory and Hyperbolic Geometry
- 3 will take Hyperbolic Geometry and Cryptography
- 15 will take Cryptography and Topology
- 8 will take Topology and Algebraic Number Theory.
Everyone else will take only one math elective. Furthermore, 16 will take either Algebraic Number Theory or Cryptography, but not both. How many math majors will take exactly one of Galois Theory, Hyperbolic Geometry, or Topology?
Answer: 17
Solution: We solve this problem by eliminating those math majors who do not fit the necessary criteria of taking exactly one of Galois Theory, Hyperbolic Geometry, or Topology. The 22 math majors who are not taking any math electives are immediately excluded. Also eliminated are the 7 taking ANT and GT, the 12 taking GT and HG, and the 3 taking HG and C, because they are not taking GT or HG exclusively. We also eliminate the 16 taking either ANT only or C only. Finally, we subtract those who are not taking Topology exclusively. Thus, we arrive at the answer to the problem
$$
100-22-7-12-3-16-15-8=17 \text {. }
$$