If the letters of the word 'MOTHER' be permuted

Question:

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__________.

Solution:

$\begin{array}{cccccc}\mathrm{M} & \mathrm{O} & \mathrm{T} & \mathrm{H} & \mathrm{E} & \mathrm{R} \\ 3 & 4 & 6 & 2 & 1 & 5\end{array}$

$\Rightarrow 2 \times 5 !+2 \times 4 !+3 \times 3 !+2 !+1$

$=240+48+18+2+1=309$