Which of the following product of e, h, $\mathrm{\mu }$, G (where $\mathrm{\mu }$ is the permeability) be taken so that the dimensions of the product are same as that of the speed of light?

Here $\mathrm{v}={\mathrm{e}}^{\mathrm{a}}{\mathrm{h}}^{\mathrm{b}}{\mathrm{\mu }}^{\mathrm{c}}{\mathrm{G}}^{\mathrm{d}}$. Taking the dimensions, ${\mathrm{M}}^0{\mathrm{LT}}^{-1}{\mathrm{A}}^0$

       $={\left[{\mathrm{AT}}^1\right]}^{\mathrm{a}}{\left[{\mathrm{ML}}^2{\mathrm{T}}^{-1}\right]}^{\mathrm{b}}{\left[{\mathrm{MLT}}^{-2}{\mathrm{A}}^{-2}\right]}^{\mathrm{c}}{\left[{\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}\right]}^{\mathrm{d}}$

There will be four simultaneous equations by equating the dimensions of M, L, T, and A. These are $\mathrm{a}-2\mathrm{c}=0, \mathrm{a}-\mathrm{b}-2\mathrm{c}-2\mathrm{d}=-1, \mathrm{b}+\mathrm{c}-\mathrm{d}=0 \text{ and }2\mathrm{b}+\mathrm{c}+3\mathrm{d}=1$

Solving for a, b, c and d, we get

        $\mathrm{a}=-2, \mathrm{b}=1, \mathrm{c}=-1, \mathrm{d}=0$

Thus, $\mathrm{v}={\mathrm{e}}^{-2}{\mathrm{h\mu }}^{-1}{\mathrm{G}}^0$