Q.If these words (formed from all letters of REMAIN) are written in dictionary order, what will be the 40th word?

Sort the 6 letters of REMAIN alphabetically: A, E, I, M, N, R. Fixing the FIRST letter as A, the remaining 5 letters can be arranged in $5!=120$ ways — since $120\ge40$, the 40th word indeed starts with A, and we need the 40th word among these 120 (i.e. the 40th arrangement of E,I,M,N,R). Fixing the second letter from the remaining sorted list E,I,M,N,R: each choice of second letter accounts for $4!=24$ words. With second letter E: words 1–24. With second letter I: words 25–48 — so the 40th word overall is the $(40-24)=16$th word among arrangements starting 'AI'. Fixing the third letter from the remaining sorted list E,M,N,R (each choice accounts for $3!=6$ words): third letter E → words 1–6 (of this block); M → words 7–12; N → words 13–18 — so the 16th word falls in the 'N' block, specifically the $(16-12)=4$th word starting ' …

Fix letters one position at a time in alphabetical order; at each step, the block size is $(\text{remaining letters})!$, and dividing the running rank by this block size (adjusting the rank as you go) identifies …

Losing track of the running rank after fixing each letter (forgetting to subtract the words already accounted for by earlier …