Note that each carbon needs exactly one double bond. Label the six carbons in the center $1,2,3,4,5,6$ clockwise. We consider how these six carbons are double-bonded. If a carbon in the center is not double-bonded to another carbon in the center, it must double-bond to the corresponding carbon on the outer ring. This will result in the outer ring broken up into (some number of) strings instead of a loop, which means that there will be at most one way to pair off the outer carbons through double-bonds. (In fact, as we will demonstrate later, there will be exactly one way.) Now we consider how many double bonds are on the center ring. - 3 bonds. There are 2 ways to pair of the six carbons, and 2 ways to pair of the outer ring as well, for 4 ways in total. - 2 bonds. Then either two adjacent carbons (6 ways) or two diametrically opposite carbons (3 ways) are not double-bonded, and in the former case the outer ring will be broken up into two "strands" with 2 and 14 carbons each, while in the latter case it will be broken up into two strands both with 8 carbons each, and each produce one valid way of double-bonding, for 9 ways in total. - 1 bond. There are 6 ways to choose the two double-bonded center carbon, and the outer ring will be broken up into four strands with $2,2,2,8$ carbons each, which gives one valid way of double-bonding, for 6 ways in total. - 0 bonds. Then the outer ring is broken up into six strands of 2 carbons each, giving 1 way. Therefore, the number of possible arrangements is $4+9+6+1=20$. Note: each arrangement of single/double bonds is also called a resonance structure of coronene.