Letters of the word ‘MOTHER’ are written in all possible ways and the words (with or without meaning) so formed are written as in a dictionary. The position of the word MOTHER is

Observe the following table:

$\text{Intial}\text{letters}$                      $\text{number}\text{of}\text{words}$

$\text{E}\times \times \times \times \times$                              ${}^5{\text{P}}_5=\overline{)5}=120$

$\text{H}\times \times \times \times \times$                              ${}^5{\text{P}}_5=\overline{)5}=120$

$\text{ME}\times \times \times \times$                              ${}^4{\text{P}}_4=\overline{)4}=24$

$\text{MH}\times \times \times \times$                             ${}^4{\text{P}}_4=\overline{)4}=24$

$\text{MOE}\times \times \times$                             ${}^3{\text{P}}_3=\overline{)3}=6$

$\text{MOH}\times \times \times$                            ${}^3{\text{P}}_3=\overline{)3}=6$

$\text{MOR}\times \times \times$                            ${}^3{\text{P}}_3=\overline{)3}=6$

$\text{MOTE}\times \times$                             ${}^2{\text{P}}_2=\overline{)2}=2$

So, MOTHER is 309 th word.

$=2\times \overline{)5}+2\overline{)4}+3\times \overline{)3}+1\times 2\overline{)2}+0\times \overline{)1}+1$