How many words can be formed by the letters of the word 'TUESDAY' which starts and ends with a vowel?

Answer

To solve the problem of how many words can be formed by the letters of the word 'TUESDAY' that start and end with a vowel, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Letters**:
The word 'TUESDAY' consists of 7 letters: T, U, E, S, D, A, Y. Among these, the vowels are U, E, and A, and the consonants are T, S, D, and Y.

2. **Determine the Positions for Vowels**:
Since we need the words to start and end with a vowel, we have:
- The first position can be filled by any of the 3 vowels (U, E, A).
- The last position can be filled by any of the remaining 2 vowels (since one vowel has already been used in the first position).

3. **Calculate the Remaining Letters**:
After placing vowels in the first and last positions, we have 5 remaining letters to arrange. These letters consist of:
- 1 remaining vowel (from the original 3 vowels, 1 is used).
- 4 consonants (T, S, D, Y).

4. **Count the Arrangements**:
The total number of arrangements of the remaining 5 letters (1 vowel + 4 consonants) can be calculated using the factorial of the number of letters:
- The number of arrangements of 5 letters is 5! (factorial of 5).

5. **Combine the Choices**:
The total number of words can be calculated by multiplying the number of choices for the first vowel, the number of choices for the last vowel, and the arrangements of the remaining letters:
\[ \text{Total Words} = (\text{Choices for 1st vowel}) \times (\text{Choices for last vowel}) \times (\text{Arrangements of remaining letters}) \]
\[ = 3 \times 2 \times 5! \]

6. **Calculate the Final Result**:
Now, we compute:
\[ 5! = 120 \]
Therefore:
\[ \text{Total Words} = 3 \times 2 \times 120 = 720 \]

### Final Answer:
The total number of words that can be formed by the letters of the word 'TUESDAY' which start and end with a vowel is **720**.

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