Step-by-step Solution:

Correct Answer: A (at least 589)

The question asks to find the position (or rank) of the word "TAILER" when all possible rearrangements of its letters are sorted in dictionary (alphabetical) order.

Here is the step-by-step method to find the rank:

1. List the letters alphabetically: The letters in "TAILER" are A, E, I, L, R, T.
2. Calculate the number of words that come before "TAILER": We do this by checking each letter of "TAILER" from left to right.
   
   • Position 1 (T): There are 5 letters smaller than 'T' (A, E, I, L, R). The number of words starting with these letters is 5 × 5! = 5 × 120 = 600.
   
   
   • Position 2 (A): With 'T' fixed, the remaining letters are {A, E, I, L, R}. There are 0 letters smaller than 'A'. So, the contribution is 0 × 4! = 0.
   
   
   • Position 3 (I): With "TA" fixed, the remaining letters are {E, I, L, R}. There is 1 letter smaller than 'I' (which is 'E'). So, the contribution is 1 × 3! = 1 × 6 = 6.
   
   
   • Position 4 (L): With "TAI" fixed, the remaining letters are {E, L, R}. There is 1 letter smaller than 'L' (which is 'E'). So, the contribution is 1 × 2! = 1 × 2 = 2.
   • Position 5 (E): With "TAIL" fixed, the remaining letters are {E, R}. There are 0 letters smaller than 'E'. So, the contribution is 0 × 1! = 0.
   
   
   • Position 6 (R): The last letter contributes 0.
3. Calculate the final rank: The number of words before "TAILER" is the sum of the values calculated: 600 + 0 + 6 + 2 + 0 + 0 = 608.
   The rank of the word "TAILER" itself is 608 + 1 = 609.

Since the rank is 609, the statement "at least 589" is true. Therefore, option A is the correct choice.