What is the probability that the resulting three-character string contains no vowels?
this is the second part to the question above please show all work.

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?
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**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.

Learn how to calculate the probability of forming a three-character string with no vowels. This problem involves counting favorable outcomes and total possible outcomes, using basic probability concepts. Suitable for students in Grades 9-12.

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