The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is:

Find the serial number of the word TOUGH when all permutations of OUGHT are arranged in dictionary order.

Listing the letters in alphabetical order gives G, H, O, T, U (positions 1, 2, 3, 4, 5).

Words starting with G: $$4! = 24$$ (positions 1-24), words starting with H: $$4! = 24$$ (positions 25-48), words starting with O: $$4! = 24$$ (positions 49-72). Words starting with T come next (positions 73 onwards).

With remaining letters G, H, O, U, words starting with TG: $$3! = 6$$ (positions 73-78), words starting with TH: $$3! = 6$$ (positions 79-84). Words starting with TO come next (position 85 onwards).

With remaining letters G, H, U, words starting with TOG: $$2! = 2$$ (positions 85-86), words starting with TOH: $$2! = 2$$ (positions 87-88). Words starting with TOU come next (position 89).

Among words starting with TOU, the remaining letters are G and H. Since G comes before H, TOUGH is the first in this list, corresponding to position 89.

The correct answer is Option A: $$89$$.