All the arrangements, with or without meaning, of the word FARMER...

Number of such words without both $R$ together

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

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

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

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

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

Number of such words start with $FARE = 2 ! = 2$

Number of such words start with $FARMER = 1$

Total $= 77$