How many of the following are aromatic compounds?

The correct answer is 5.

To determine how many of the given compounds are aromatic, we apply Hückel's Rule of Aromaticity. According to Hückel's rule, a cyclic, planar, and fully conjugated molecule is classified as:
1. Aromatic if it possesses $4n+2$ delocalized π-electrons (where $n=0,1,2,3,...$).
2. Antiaromatic if it possesses $4n$ delocalized π-electrons (where $n=1,2,3,...$).

Let us analyze each of the six compounds visible in the provided image from left to right, top to bottom:

1. Naphthalene (First compound in the top row):
This molecule consists of two fused benzene rings. It is planar, cyclic, and fully conjugated. The total number of conjugated π-electrons is 10. Since
$10=4\left(2\right)+2$
(which satisfies the $4n+2$ rule with $n=2$), naphthalene is aromatic.

2. Fluorenyl Anion (Second compound in the top row):
This structure consists of two benzene rings fused to a central five-membered ring containing a negative charge (lone pair). The negative charge resides on a carbon atom that undergoes hybridization change from sp³ to sp² to allow complete delocalization throughout the cyclic system. The total number of conjugated π-electrons is 14 (12 from the double bonds and 2 from the carbanion). Since
$14=4\left(3\right)+2$
(which satisfies the $4n+2$ rule with $n=3$), the fluorenyl anion is aromatic.

3. Diphenylmethane (Third compound in the top row):
This molecule contains two benzene rings linked by a non-conjugated sp³ methylene (-CH₂-) bridge. Since the two benzene rings themselves are independently aromatic (each having a planar cyclic system with 6 π-electrons, satisfying $4n+2$ for $n=1$), the compound contains aromatic rings and is classified as aromatic.

4. Cyclohexenylbenzene (First compound in the bottom row):
This compound features a benzene ring substituted with a cyclohexene ring. The benzene ring remains intact as a fully conjugated cyclic system containing 6 π-electrons. Since it contains a benzene ring, the compound is aromatic.

5. Dihydronaphthalenyl Cation derivative (Second compound in the bottom row):
This molecule contains a benzene ring fused to a partially unsaturated cationic six-membered ring. The benzene portion of the molecule contains 6 π-electrons in a cyclic conjugated system, making the compound aromatic.

6. Cycloheptatrienyl Anion (Third compound in the bottom row):
This is a seven-membered ring with three conjugated double bonds and a negative charge (lone pair). The total number of π-electrons participating in conjugation is:
$6\text{(from three double bonds)}+2\text{(from the negative charge)}=8\text{π-electrons}$.
Since 8 is a multiple of 4 (matching the $4n$ rule for $n=2$), this fully conjugated planar system is antiaromatic.

Conclusion:
Out of the 6 compounds, 5 are aromatic, and 1 (cycloheptatrienyl anion) is antiaromatic.