Identify given parameters

The radius of the large solid metallic sphere is $R = 12\text{ cm}$ and the radius of each small sphere is $r = 2\text{ cm}$.

$R = 12\text{ cm}, \quad r = 2\text{ cm}$

Set up volume conservation equation

When a solid sphere is melted and recast, the total volume remains constant. Thus, the volume of the large sphere equals $n$ times the volume of one small sphere: $V_{\text{large}} = n \times V_{\text{small}}$.

$\frac{4}{3}\pi R^3 = n \times \frac{4}{3}\pi r^3$

Simplify and solve for n

Cancelling out $\frac{4}{3}\pi$ from both sides, we get $n = \left(\frac{R}{r}\right)^3$. Substituting the values: $n = \left(\frac{12}{2}\right)^3 = 6^3 = 216$.

$n = 6^3 = 216$