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We want exactly 2 out of the 4 selected inventions to be from the 19th century, and the other 2 from different centuries.
Number of ways to choose 2 from the 3 19th-century inventions:
Number of ways to choose 2 from the 7 non-19th-century inventions:
Total ways to choose any 4 inventions out of 10:
First, compute the total number of unrestricted arrangements of 6 distinct breakthroughs:
Number of ways to arrange these 5 entities:
Subtract from total to get non-adjacent arrangements:
We use inclusion-exclusion to compute the number of assignments where all 3 bee types are used at least once.
Total number of assignments: each of 9 plants gets one of 3 bee types:
By inclusion-exclusion, number of assignments missing at least one type: