To count the number of strings over the alphabet {a, b, c} that begin with the letter b and have a length of 7 or 8, we can break this down into two cases:
Case 1: Strings of length 7 starting with 'b'
In this case, we have 7 positions to fill with the letters 'a', 'b', or 'c'. The first position is fixed as 'b', and the remaining 6 positions can each be filled with one of the three letters.
So, for this case, there are 3 choices for each of the 6 remaining positions:
3^6 = 729 strings
Case 2: Strings of length 8 starting with 'b'
In this case, we have 8 positions to fill with the letters 'a', 'b', or 'c'. The first position is fixed as 'b', and the remaining 7 positions can each be filled with one of the three letters.
So, for this case, there are 3 choices for each of the 7 remaining positions:
3^7 = 2187 strings
Now, add the results from both cases:
729 (from Case 1) + 2187 (from Case 2) = 2916 strings
So, there are 2916 strings over the alphabet {a, b, c} that begin with the letter 'b' and have a length of 7 or 8.
The closest option to this count is d) 37, which is not correct. The correct answer is not among the given options.