Final answer:
There are 216 5-letter strings made up of the digits a, b, and c that do not include the string "aaa". This is calculated by finding the total number of possible strings (243) and subtracting the ones that contain "aaa" (27). The correct option is B.
Step-by-step explanation:
To solve how many 5-letter strings can be made up with the digits a, b, and c that do not include the string "aaa", we can first calculate the total number of 5-letter strings without restriction and then subtract the number of strings that do contain "aaa".
Without restriction, each position in the 5-letter string can be occupied by any of the three digits, giving us a total of 3³³ = 243 possible strings. Now, we need to consider the strings that include "aaa" as a consecutive sequence. This sequence can appear at the beginning (a XY), in the middle (XaaaY), or at the end (XYaaa), where X and Y represent any of the three digits. For each of these cases, there are 3² = 9 possibilities, because X and Y can each be a, b, or c. So, 3 * 9 = 27 strings include "aaa".
To find the answer, we subtract the number of strings containing "aaa" from the total possible strings: 243 - 27 = 216. Therefore, there are 216 5-letter strings made up of the digits a, b, c that do not include the string "aaa".