To find the number of words that can be formed using all letters of the word "EQUATION" without repetition, we need to consider the following:
1. Total number of letters in the word: EQUATION has 8 letters.
2. Number of ways to arrange these letters: Since we are using all letters without repetition, the number of ways to arrange them is 8!.

Now, let's break down the solution step by step:

Step 1: Counting the number of ways to arrange the letters

The word "EQUATION" has 8 letters, so we have 8 choices for the first letter, 7 choices for the second letter, 6 choices for the third letter, and so on. This can be calculated as follows:

8 choices for the first letter
7 choices for the second letter
6 choices for the third letter
5 choices for the fourth letter
4 choices for the fifth letter
3 choices for the sixth letter
2 choices for the seventh letter
1 choice for the eighth letter

Using the counting principle, we can multiply these choices together to get the total number of arrangements:

8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 8!

Step 2: Considering the repetition of vowels

In the word "EQUATION," there are two vowels: E and A. Since these vowels are repeated, we need to account for the overcounting that occurs when we treat them as distinct letters.

To do this, we divide the total number of arrangements by the number of ways the vowels can be arranged among themselves. In this case, there are 2 vowels, so the number of ways to arrange them is 2!.

Therefore, the adjusted total number of arrangements is:

8! / 2!

Step 3: Evaluating the factorial expressions

To simplify the expression, we calculate the factorials:

8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
2! = 2 x 1 = 2

Substituting these values into the adjusted expression:

8! / 2! = 40,320 / 2 = 20,160

Therefore, the number of words that can be formed using all letters of the word "EQUATION" is 20,160.

None of the given answer options match this result. Thus, it seems that an error occurred in the options provided.