What will be the 55th word in the arrangement of the letters of the word PERFECT

Concept

To rank a word formed from given letters in dictionary (alphabetical) order, sort the letters first. At each position, count how many arrangements of the remaining letters would start with an earlier letter than the one actually chosen — for a set of k remaining letters with a letter repeated r times, that count is k!/r!. Adding up these counts position by position pins down the exact word at any target rank.

Working

1. Sort the letters of PERFECT alphabetically: C, E, E, F, P, R, T (7 letters, with E repeated twice).

2. Fix the 1st letter as C — the remaining 6 letters (E, E, F, P, R, T) arrange in 6!/2! = 360 ways, already more than 55, so the word begins with C.

3. Fix the 2nd letter as E — the remaining 5 letters (E, F, P, R, T) are now all distinct and arrange in 5! = 120 ways, still more than 55, so the word begins CE.

4. For the 3rd letter, split by the remaining letters E, F, P, R, T; each block fixes 4 distinct letters that arrange in 4! = 24 ways: CEE-block covers positions 1-24, CEF-block covers positions 25-48, CEP-block covers positions 49-72. Since 48 < 55 <= 72, the 3rd letter is P, and the target is the (55 - 48) = 7th word inside the CEP-block.

5. For the 4th letter, split the CEP-block by the remaining letters E, F, R, T; each choice fixes 3 distinct letters that arrange in 3! = 6 ways: choosing E covers positions 1-6, choosing F covers positions 7-12 of this block. The target position 7 falls under the F-choice, so the 4th letter is F, and the target is the (7 - 6) = 1st word inside that sub-block.

6. The remaining letters E, R, T are already distinct; their 1st arrangement in sorted order is simply E, R, T.

7. Concatenating every letter fixed at each stage - C, E, P, F, E, R, T - gives the word CEPFERT.

Cross-check

Cross-check: the words before this arrangement number 24 (CEE-block) + 24 (CEF-block) + 6 (CEPE-block) = 54, so CEPFERT is exactly the 55th word - confirming the count independently.

So the 55th word in the dictionary arrangement of the letters of PERFECT is CEPFERT.