To determine the probability that the resulting three-character string contains no vowels, we first need to identify the vowels in each word and then count the number of outcomes that do not include any vowels.

Step-by-Step Solution

1. Identify the vowels in each word:

   • "TACO": vowels are A and O
   • "MATH": vowel is A
   • "DOG": vowel is O

2. Identify the consonants in each word:

   • "TACO": consonants are T and C (2 choices)
   • "MATH": consonants are M, T, and H (3 choices)
   • "DOG": consonants are D and G (2 choices)

3. Calculate the number of possible outcomes that contain no vowels:

   • For the first character (from "TACO"), there are 2 choices (T, C).
   • For the second character (from "MATH"), there are 3 choices (M, T, H).
   • For the third character (from "DOG"), there are 2 choices (D, G).

   The total number of no-vowel outcomes is: $2 \times 3 \times 2 = 12$

4. Calculate the total number of possible outcomes (from part a):

   • We previously calculated that the total number of possible outcomes is 48.

5. Determine the probability of a no-vowel outcome: $\text{Probability} = \frac{\text{Number of no-vowel outcomes}}{\text{Total number of possible outcomes}} = \frac{12}{48} = \frac{1}{4}$

Thus, the probability that the resulting three-character string contains no vowels is $\boxed{\frac{1}{4}}$.

Would you like more details or have any questions?

Here are 5 questions you can ask next:

1. How do you calculate the probability of events in a sample space?
2. Can you explain the concept of independent and dependent events in probability?
3. What are some examples of combinatorial problems in probability?
4. How do you use the principle of inclusion-exclusion in probability?
5. What is the difference between theoretical and experimental probability?

Tip: When dealing with probabilities, ensure that you correctly count the favorable outcomes and the total possible outcomes, and then express the probability as a ratio or fraction.