In how many arrangements of the letters of BANANA are the two N's not next to each other?

Arrange B, A, A, A first: $\frac{4!}{3!}=4$ ways. They leave 5 gaps; put the 2 identical N's in 2 different gaps: ${}^5{C}_2=10.4\times 10=40$. (Check: total 60, N's together 20.)