To solve the problem step by step, we will break it into two parts as stated in the question.
### Part 1: Total Number of Three-Letter Code Words
1. **Identify the Total Number of Cards**: We have 7 cards, each bearing a letter from the word "DOUBLES".
2. **Choose the First Letter**: For the first position in the three-letter code word, we can choose any of the 7 letters. So, there are 7 possible choices.
3. **Choose the Second Letter**: After selecting the first letter, we have 6 letters remaining. Thus, there are 6 possible choices for the second position.
4. **Choose the Third Letter**: After selecting the first and second letters, we have 5 letters left. Therefore, there are 5 possible choices for the third position.
5. **Calculate the Total Combinations**: The total number of three-letter code words can be calculated by multiplying the number of choices for each position:
\[
\text{Total Code Words} = 7 \times 6 \times 5 = 210
\]
### Part 2: Code Words with a Vowel Between Two Consonants
1. **Identify Vowels and Consonants**: In the word "DOUBLES", the vowels are O, U, and E (3 vowels), and the consonants are D, B, L, and S (4 consonants).
2. **Positioning**: We need to form a three-letter code word where the structure is Consonant - Vowel - Consonant (CVC).
3. **Choose the First Consonant**: For the first position (C), we can choose any of the 4 consonants. So, there are 4 choices.
4. **Choose the Vowel**: For the second position (V), we can choose any of the 3 vowels. Thus, there are 3 choices.
5. **Choose the Second Consonant**: For the third position (C), we can choose from the remaining 3 consonants (since one consonant has already been used in the first position). Therefore, there are 3 choices.
6. **Calculate the Total Combinations for CVC Structure**: The total number of code words that consist of a vowel between two consonants can be calculated by multiplying the number of choices for each position:
\[
\text{Total CVC Code Words} = 4 \times 3 \times 3 = 36
\]
### Final Answers:
- The total number of three-letter code words that can be formed is **210**.
- The number of these words that consist of a vowel between two consonants is **36**.
