Words From Dictionary Rank

Finding the rank of a word in dictionary order counts how many arrangements precede it by fixing earlier letters alphabetically, a classic JEE Advanced permutation problem. The letters of RANK in alphabetical order are A, K, N, R. To rank RANK, count words starting with a letter earlier than R: words starting with A, K, or N each contribute 3! = 6 arrangements of the remaining three letters, giving 3 × 6 = 18 words before any word starting with R. Within words starting with R, the remaining letters are A, K, N; RANK takes A next (the smallest), so no skip occurs at the second position. At the third position the remaining letters are K and N, and RANK takes N, so the one word with K there, namely RAKN, precedes RANK, contributing 1 more. Total words preceding RANK = 18 + 1 = 19, so its rank is 19 + 1 = 20. Option 18 forgets to count RANK itself. Option 16 undercounts the preceding prefixes. Option 22 overcounts within the R-block. Hence the rank is 20. Plausibility check: there are 4! = 24 total arrangements, and RANK is near but not at the end, so a rank of 20 out of 24 is plausible and consistent with the letter-by-letter count.