The letters of the word OUGHT are written in all possible ways and these words are arranged as in a…

Lets arrange the letters of OUGHT in alphabetical order.

$\mathrm{G},\mathrm{H},\mathrm{O},\mathrm{T},\mathrm{U}$

Number of words starting with $G$

$=4!=24$

Number of words starting with $H$

$=4!=24$

Number of words starting with  $O$

$=4!=24$

Number of words starting with $TG$

$3!=6!$

Number of words starting with $TH$

$=3!=6$

Number of words starting with $TOG$

$=2!=2$

Number of words starting with $TOH$

$=2!=2$

Next word is $\mathrm{TOUGH}$.

Hence, the serial number of the word $\mathrm{TOUGH}$ is

$$\begin{gathered} =24+24+24+6+6+2+2+1 \\ =89 \end{gathered}$$