The question asks to identify the correct floral formula for the Solanaceae family (commonly known as the potato family) among the given options.

Floral Formula Breakdown

A floral formula represents the symmetry, sexuality, and whorl characteristics of a flower using standardized botanical symbols:

• Symmetry: $\oplus$ (Actinomorphic / Radial symmetry)
• Sexuality: $\text{\textdied}$ or $\stackrel{\textstyle\circ}{+}$ (Bisexual / Hermaphrodite)
• Calyx (Sepals): $K$. The number indicates the count, and parentheses $( )$ denote cohesion/fusion (gamosepalous).
• Corolla (Petals): $C$. The number indicates the count, and parentheses $( )$ denote cohesion/fusion (gamopetalous).
• Androecium (Stamens): $A$. The number indicates the count.
• Adhesion Tie / Arc: An overarching tie or arc bridging $C$ and $A$ ($\overparen{C\ A}$) indicates epipetalous stamens (stamens fused/inserted onto the petals).
• Gynoecium (Carpels): $G$. The number indicates the count, parentheses $( )$ denote syncarpy (fused carpels), and an underline ($\underline{G}$) denotes a superior ovary.

Solanaceae Family Characteristics

Flowers of the Solanaceae family typically exhibit the following diagnostic features:

• Symmetry: Actinomorphic ($\oplus$).
• Sexuality: Bisexual ($\text{\textdied}$).
• Calyx: 5 sepals, united / gamosepalous, persistent ($K_{(5)}$).
• Corolla: 5 petals, united / gamopetalous, typically with valvate aestivation ($C_{(5)}$).
• Androecium: 5 stamens, polyandrous (free from each other), and epipetalous (adherent to the corolla tube). This adhesion is formally represented by the connecting tie: $\overparen{C_{(5)}\ A_5}$.
• Gynoecium: Bicarpellary, syncarpous (fused), with an obliquely placed superior ovary ($\underline{G}_{(2)}$).

Evaluating the Options

• Option 1 ($\oplus\ \text{\textdied}\ K_{(5)}\ C_{(5)}\ A_5\ \underline{G}_{(2)}$): Incorrect. It fails to include the tie/arc between $C$ and $A$, incorrectly implying that the stamens are free from the corolla rather than epipetalous.
• Option 2 ($\oplus\ \text{\textdied}\ K_5\ C_{(5)}\ A_5\ \underline{G}_{(2)}$): Incorrect. Shows free sepals ($K_5$) instead of fused sepals ($K_{(5)}$) and omits the epipetalous arc.
• Option 3 ($\oplus\ \text{\textdied}\ K_5\ C_5\ A_5\ \underline{G}_{(2)}$): Incorrect. Erroneously denotes both sepals ($K_5$) and petals ($C_5$) as free.
• Option 4 ($\oplus\ \text{\textdied}\ K_{(5)}\ \overparen{C_{(5)}\ A_5}\ \underline{G}_{(2)}$): Correct. Accurately represents the actinomorphic symmetry, bisexual nature, gamosepalous calyx, gamopetalous corolla, epipetalous stamens (via the connecting arc bridging $C$ and $A$), and a bicarpellary, syncarpous, superior ovary.

Conclusion

The correct floral formula for the Solanaceae family is:

$$\oplus\ \text{\textdied}\ K_{(5)}\ \overparen{C_{(5)}\ A_5}\ \underline{G}_{(2)}$$

Therefore, Option 1 is the correct option.