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How many strings of length 10 over the alphabet {a, b, c} have either exactly three a's or exactly four b's?

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There are 10 places to be filled. 

Case 1: Exactly 3 a's: We fill 3 places with 3 a's in 3C3 * 10C3 ways. Now, remaining 7 places can be filled using b or c in 27 ways. 

Case 2: Exactly 4 b's: We fill 4 places with 4 b's in 4C4 * 10C4 ways. Remaining 6 places can be filled using a or c in 26 ways.

Case 3: Exactly 3 a's and exactly 4 b's: 10C3 * 7C4 * 13 = 4200

So, number of strings possible = Case 1 + Case 2 - Case 3 = 10C3 * 128 + 10C4 * 64 - 4200 = 24,600 

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