How many different words can be formed using the letters of the word “EDUCATION‘ such that , the word always ends with a vowel?

To solve this problem, we need to consider two things: the number of ways we can arrange the letters in the word "EDUCATION", and the requirement that the word must end with a vowel.

1. Count the number of letters and identify the vowels and consonants: The word "EDUCATION" has 9 letters, 5 of which are vowels (E, U, A, I, O) and 4 are consonants (D, C, T, N).

2. Calculate the number of arrangements: The total number of arrangements of the word "EDUCATION" without any restrictions is 9! (9 factorial), which is 362,880. However, we have to account for the repeated letter 'U'. So, we divide 9! by 2! to get 181,440.

3. Adjust for the requirement that the word must end with a vowel: Since the word must end with a vowel, we consider one of the 5 vowels to be fixed at the end. This leaves us with 8 letters to arrange, which can be done in 8! (8 factorial) ways, or 40,320. But again, we have to account for the repeated letter 'U', so we divide 8! by 2! to get 20,160.

4. Multiply by the number of choices for the final vowel: Since any of the 5 vowels can be at the end, we multiply the number of arrangements by 5. So, 20,160 * 5 = 100,800.

So, there are 100,800 different words that can be formed using the letters of the word "EDUCATION" such that the word always ends with a vowel.