Question

All the arrangements, with or without meaning, of the word $FARMER$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $FARMER$ in this list is

Answer 1

First find all possible words and then subtract words from each case that have both R together i.e.,

$A.....\Rightarrow \frac{5!}{2!}-4!=36$
$E.....\Rightarrow \frac{5!}{2!}-4!=36$
$FAE.....\Rightarrow \frac{3!}{2!}-2=1$
$FAM.....\Rightarrow \frac{3!}{2!}-2=1$
$FARE.....\Rightarrow 2!=2$
$FARMER.....\Rightarrow 1=1$

$\therefore$ Rank of farmer is $77.$