All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has…

Number of such words without both $R$ together

$=$ total possible words - total possible words with both $R$ together

Number of such words start with $A=\frac{5!}{2!}-4!=36$

Number of such words start with $E=\frac{5!}{2!}-4!=36$

Number of such words start with $\mathrm{FAE}=\frac{3!}{2!}-2=1$

Number of such words start with $\mathrm{FAM}=\frac{3!}{2!}-2=1$

Number of such words start with $\mathrm{FARE}=2!=2$

Number of such words start with $\mathrm{FARMER}=1$

Total $=77$