To find the number of words that can be formed by using the letters of the word "EQUATION" if each word has to start with a vowel, we can use permutation.
First, let's identify the vowels in the word "EQUATION," which are E, U, and A. Since each word has to start with a vowel, we need to fix the position of the first letter as one of these three vowels.
To solve this problem, we'll break it down into cases:
1. If the first letter is E: We have to arrange the remaining letters U, A, Q, T, I, O and N. So we have 7 letters to arrange, which can be done in 7! ways (7 factorial).
2. If the first letter is U: We have to arrange the remaining letters E, A, Q, T, I, O, and N. Again, we have 7 letters to arrange, which can be done in 7! ways.
3. If the first letter is A: We have to arrange the remaining letters E, U, Q, T, I, O, and N. Once more, we have 7 letters to arrange, which can be done in 7! ways.
Since the cases are mutually exclusive (we can't have more than one first letter), we need to sum up the number of arrangements in each case:
Total = Number of arrangements if first letter is E + Number of arrangements if first letter is U + Number of arrangements if first letter is A
Total = 7! + 7! + 7!
Calculating this, we get:
Total = 5040 + 5040 + 5040
Total = 15120
Therefore, the number of words that can be formed by using the letters of the word "EQUATION" if each word has to start with a vowel is 15,120.