To find the rank of the word FATHER, we need to consider all the words that can be formed by rearranging the letters of FATHER in alphabetical order.

Step 1: Arrange the letters in alphabetical order
The letters in FATHER are already arranged in alphabetical order, so we can move on to the next step.

Step 2: Find the words starting with each letter
We will consider the words starting with each letter of FATHER. Since the first letter is 'A', we need to find the rank of all the words starting with 'A' first.

Step 3: Find the rank of words starting with 'A'
The remaining letters after removing 'A' from FATHER are F, H, T, E, and R. We can arrange these letters in (5!) = 120 ways. But we need to consider that some of these arrangements may have the same letters repeated.

Step 4: Find the number of arrangements with repeated letters
In the word FATHER, the letters F and R are repeated. To find the number of arrangements with repeated letters, we divide the total number of arrangements by the factorial of the number of repetitions. In this case, we have 2 repetitions of F and 2 repetitions of R.

Number of arrangements with repeated letters = 5! / (2! * 2!) = 5 * 4 * 3 * 2 * 1 / (2 * 1 * 2 * 1) = 5 * 2 * 3 = 30

Step 5: Find the rank of words starting with 'A'
To find the rank of words starting with 'A', we need to consider all the words starting with 'A' that can be formed by rearranging the remaining letters (F, H, T, E, R) in alphabetical order.

The words starting with 'A' in alphabetical order are:
AFTER, ARTHE, EARTH, FERTH, FRATE, HEART, RAFTHE, and REATH.

Since 'FATHER' is the 3rd word in alphabetical order among the words starting with 'A', the rank of the word FATHER would be 3.

Therefore, the correct answer is option C) 261.