If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the work Mother.

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Solution

Order is E, H, M, O, R, T
words with E _ _ _ _ _ $= 5! = 120$
words with H _ _ _ _ _ $= 5! = 120$
words with ME _ _ _ _ $= 4! = 24$
words with MH _ _ _ _ $= 4! = 24$
words with MOE _ _ _ $= 3! = 6$
words with MOH _ _ _ $= 3! = 6$
words with MOR _ _ _ $= 3! = 6$
words with MOTE_ _ $= 2! = 2$
___________
words with MOTHER $= 308$
$\Rightarrow 309^{t h}$ word.

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If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the work Mother.

Order is E, H, M, O, R, Twords with E _ _ _ _ _=5!=120words with H _ _ _ _ _=5!=120words with ME _ _ _ _=4!=24words with MH _ _ _ _=4!=24words with MOE _ _ _=3!=6words with MOH _ _ _=3!=6words with MOR _ _ _=3!=6words with MOTE_ _=2!=2 ___________words with MOTHER=308⇒309th word.