Words From Dictionary Rank

Question

If all the distinct arrangements of the letters of the word RANK are listed in dictionary order, then the position or rank of the word RANK in that ordered list equals which number?

RankGuru · Accepted Answer

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.

Words From Dictionary Rank

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