The number of ways of arranging letters of the word HAVANA so that $\mathrm{V}$ and $\mathrm{N}$ do not appear together is

We can arrange the letters $\mathrm{H}, \mathrm{A}, \mathrm{A}, \mathrm{A}$ in $\frac{4 !}{3 !}=4$ ways. If one possible arrangement is $$ X X X X $$ Then, we can arrange $\mathrm{V}, \mathrm{N}$ at any of the two places marked with $\mathrm{O}$ in the following arrangement $$ \mathrm{O} \times \mathrm{O} \times \mathrm{O} \times \mathrm{O} \times \mathrm{O} $$ Thus, we can arrange $V$ and $N$ in ${ }^5 P_2=20$ ways. Thus, the number of ways in which letters can be arranged is $4 \times 20=80$