The letters of the word OUGHT are written in all possible ways and...

Lets arrange the letters of OUGHT in alphabetical order.

$G, H, O, T, U$

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 $TOUGH$.

Hence, the serial number of the word $TOUGH$ is

$= 24 + 24 + 24 + 6 + 6 + 2 + 2 + 1 = 89$