Final answer:
The number of possible five-letter strings using the letters a, b, c, and d with repetitions allowed is 4^5. However, the number of such strings that do not contain the substring 'bad' takes a little more calculation and the exact figure must be found by subtracting the combinations that contain 'bad' from the total possible combinations of 4^5
Step-by-step explanation:
To find out how many five-letter strings can be formed using the letters a, b, c, and d, with repetitions allowed, we calculate 4 possibilities for each position. Since there are five positions, the total number of five-letter strings is 4^5.
For the second part, we use the complement. First, we find how many strings include the substring 'bad'. This substring can appear at the beginning, middle, or end, giving us three distinct cases:
- 'bad__' - The last two positions can be any of the 4 letters. So, 4 x 4 possibilities.
- '_bad_' - The first and last positions can be any of the 4 letters. Similarly, 4 x 4 possibilities.
- '__bad' - The first two positions can be any of the 4 letters. Again, 4 x 4 possibilities.
However, this overcounts strings where 'bad' might start in the second and fourth positions simultaneously, like 'abadb'. We should subtract these overcounted cases, which amount to 4 possibilities (one for each letter that can start the string).
So the total count of strings containing 'bad' is 3 x (4 x 4) - 4. To find the number of strings without 'bad', we subtract this from the total:
4^5 - [3 x (4 x 4) - 4]
Finally, we find that option (a) 4^5; 3^5 is closest to our calculations but is slightly incorrect for the second number. The exact answer isn't provided as an option, but we use this method to arrive at the correct answer.