If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is \_\_\_\_\_\_\_.
Answer: 309
Solutions · 1
Official solution
Alphabetical positions of MOTHER's letters (E,H,M,O,R,T): M is 3rd, O is 4th, T is 6th, H is 2nd, E is 1st, R is 5th.
Words starting with E: 5!=120; H: 5!=48? Let's use standard rank algorithm: for each letter position, count letters alphabetically smaller than the actual letter among remaining, times factorial of remaining positions.
Position 1 (M, rank 3 among E,H,M,O,R,T): letters smaller are E,H (2 letters) →2×5!=240
Position 2 (O, remaining A,E,H,O,R,T... remaining after M: E,H,O,R,T; O is rank 3): smaller are E,H (2) →2×4!=48
Position 3 (T, remaining E,H,R,T; T is rank 4): smaller are E,H,R (3) →3×3!=18
Position 4 (H, remaining E,H,R; H is rank 2): smaller is E (1) →1×2!=2
Position 5 (E, remaining E,R; E is rank1): smaller: none →0
Position 6 (R, remaining R): 0
\[\text{Rank} = 240+48+18+2+0+0+1 = 309\]