How many words (with or without meaning) can be formed using all letters of the word EQUATION, using each letter exactly once?

Options

A. 38320
B. 39320
C. 20160
D. None of these

Answer

To find out how many words can be formed using all the letters of the word "EQUATION," we can follow these steps:

### Step 1: Count the total number of letters
The word "EQUATION" consists of 8 letters: E, Q, U, A, T, I, O, N.

### Step 2: Check for repeated letters
Next, we need to check if any letters are repeated. In this case, all the letters in "EQUATION" are unique. Therefore, there are no repeated letters.

### Step 3: Calculate the factorial of the number of letters
Since all the letters are unique, the total number of different arrangements (or permutations) of these letters can be calculated using the factorial of the number of letters. The formula for permutations of n distinct objects is given by n! (n factorial).

Here, n = 8 (the number of letters in "EQUATION"). Thus, we calculate:

\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]

### Step 4: Perform the calculation
Now, let's compute the value of 8! step by step:

- \(8 \times 7 = 56\)
- \(56 \times 6 = 336\)
- \(336 \times 5 = 1680\)
- \(1680 \times 4 = 6720\)
- \(6720 \times 3 = 20160\)
- \(20160 \times 2 = 40320\)
- \(40320 \times 1 = 40320\)

Thus, \(8! = 40320\).

### Final Answer
The total number of words (with or without meaning) that can be formed using all the letters of the word "EQUATION" is **40,320**.

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