How many distinct arrangements of the letters of the word BANANA keep the two N's always together?

Tie the two N's into one block. The units are then the block, B, A, A, A — 5 units with A repeated 3 times, giving 5!/3! = 120/6 = 20. (The two N's are identical, so the block has no internal rearrangement. The total number of arrangements without the restriction is 6!/(3! $x$ 2!) = 60.)