According to Newton's law of universal gravitation, the force $F$ between two masses $M_1$ and $M_2$ separated by a distance $r$ is given by $F = \frac{GM_1M_2}{r^2}$. Rearranging this equation for $G$, we get $G = \frac{Fr^2}{M_1M_2}$. The dimensional formula for force $F$ is $[MLT^{-2}]$. The dimension for distance $r$ is $[L]$, so $r^2$ has dimensions $[L^2]$. The dimension for mass $M$ is $[M]$, so $M_1M_2$ has dimensions $[M^2]$. Substituting these into the formula for $G$: $[G] = \frac{[MLT^{-2}][L^2]}{[M^2]} = [M^{1-2}L^{1+2}T^{-2}] = [M^{-1}L^3T^{-2}]$.