In how many different ways can the letters of the word OPERATE be arranged?

The word OPERATE has 7 letters:

O - P - E - R - A - T - E

Counting the frequency of each letter:

• O → 1 time
• P → 1 time
• E → 2 times
• R → 1 time
• A → 1 time
• T → 1 time

The letter E appears twice, while all other letters appear once.

For permutations with repetition, the formula is:

$\text{Number of arrangements} = \frac{n!}{r_1! \times r_2! \times ...}$

where $n$ is the total number of letters and $r_1, r_2, ...$ are the frequencies of repeated letters.

For OPERATE:

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

$= \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1}$

$= \frac{5040}{2}$

$= 2520$

Therefore, the letters of the word OPERATE can be arranged in 2520 different ways.