Final answer:
To find the number of unique permutations of the letters in the word ADVANCED, use the multiset permutation formula considering the repetition of letters, resulting in 10,080 unique permutations.
Step-by-step explanation:
The number of unique permutations of the letters in the word ADVANCED can be determined using the formula for permutations of a multiset.
Since there are repeated letters, we cannot simply use the factorial of the number of letters.
There are 8 letters in total with 1 'A', 2 'D's, 1 'V', 2 'N's, 1 'C', and 1 'E'.
The formula for permutations of a multiset is
P = \(\dfrac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_r!}\)
Where n is the total number of items, and n_1, n_2, \ldots, n_r are the frequencies of each unique item.
In this case, it's:
P = \(\dfrac{8!}{1! \cdot 2! \cdot 1! \cdot 2! \cdot 1! \cdot 1!}\) = \(\dfrac{40,320}{4}\) = 10,080.
Therefore, there are 10,080 unique permutations of the letters in the word ADVANCED.