Problem of the Week Problem D No Power

Five balls are placed in a bag. Each ball is labelled with a $$2$$, $$4$$, $$6$$, $$8$$, or $$10$$, with no ball having the same label as any other. Adeleke randomly chooses a ball, records the integer on the ball, and replaces the ball into the bag. Then Bo randomly chooses a ball, records the integer on the ball, and replaces the ball into the bag. Finally, Carlos randomly chooses a ball, records the integer on the ball, and replaces the ball into the bag.

Determine the probability that the product of the three recorded integers is not a power of $$2$$.