How many words beginning with vowels can be formed with the letters of the word EQUATION?

Options

A. 25200
B. 15200
C. 25300
D. 35200

Answer

To find the number of words beginning with vowels that can be formed using the letters of the word "EQUATION", we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Vowels**:
The vowels in the word "EQUATION" are E, U, A, I, and O. Thus, we have a total of 5 vowels.

2. **Total Letters**:
The word "EQUATION" consists of 8 letters in total.

3. **Fix the First Letter**:
Since we want the words to begin with a vowel, we can choose any one of the 5 vowels to be the first letter.

4. **Count Remaining Letters**:
After choosing a vowel for the first position, we have 7 letters left (since one vowel is already used).

5. **Calculate Arrangements**:
The remaining 7 letters can be arranged in any order. The number of arrangements of 7 letters is given by \(7!\) (7 factorial).

6. **Calculate Total Words**:
Since we have 5 choices for the first letter and \(7!\) arrangements for the remaining letters, the total number of words beginning with a vowel is:
\[ \text{Total Words} = 5 \times 7! \]

7. **Calculate \(7!\)**:
\(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\).

8. **Final Calculation**:
Now, substituting \(7!\) into the total words formula:
\[ \text{Total Words} = 5 \times 5040 = 25200. \]

### Conclusion:
Thus, the total number of words that can be formed with the letters of the word "EQUATION" that begin with a vowel is **25200**.

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