In how many different ways can the letters of the word OPERATE be arr…

The correct answer is Option (2) → 2520

Step 1: Count the total number of letters

The word OPERATE has 7 letters in total ($n = 7$).

Step 2: Identify repeating letters

Let's look at the frequency of each letter:

• O: 1
• P: 1
• E: 2 (repeats twice)
• R: 1
• A: 1
• T: 1

Step 3: Apply the Permutation Formula

The number of arrangements is given by:

$\frac{n!}{n_1! \times n_2! \times \dots \times n_k!}$

Where $n$ is the total number of letters and $n_k$ are the counts of repeating letters.

$\text{Number of ways} = \frac{7!}{2!}$

Step 4: Calculate the value

• $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = \mathbf{5040}$
• $2! = 2 \times 1 = \mathbf{2}$

$\text{Total arrangements} = \frac{5040}{2} = \mathbf{2520}$