it's been a while since i had permutations class, please correct me if i'm wrong

A C T I V A T E D

if 2 A's are separated by 5 letters then the word would be in the form: A (5 random letters) A (2 random letters)

If there were no repeated letters, the number of permutations would be 7! for 7 blank spaces

Since T is present twice, we divide it by 2!, which would mean number of permutations = 7!/2! = 2520

Similarly, the two A's can be separated by 6 and 7 letters too, total permutations = 2520 * 3 = 7560

forgive me if i am wrong i tried :)