What will be the rank of word OPTION in the dictionary of words formed by…

Key idea: find how many distinct words come before "OPTION" in lexicographic order. For each position, count choices of a smaller letter and multiply by permutations of the remaining letters (remember there are two O's, so divide by 2! when both O's remain).

• Total distinct permutations of letters of OPTION = 6!/2! = 360 (two O's).

• First letter smaller than 'O': choices I or N → for each choice permutations = 5!/2! = 60. Subtotal = 2 × 60 = 120.

• First letter O, second letter smaller than 'P': choices I, N, or O → each gives 4! = 24. Subtotal = 3 × 24 = 72.

• First two letters O P, third letter smaller than 'T': choices I, N, or O → each gives 3! = 6. Subtotal = 3 × 6 = 18.

• Fourth letter is I: no smaller remaining letter → subtotal = 0.

• Fifth letter is O: among remaining letters (N,O), the letter N is smaller → subtotal = 1 × 1! = 1.

Total words that come before "OPTION" = 120 + 72 + 18 + 0 + 1 = 211. Therefore the rank of "OPTION" = 211 + 1 = 212.