a
Time period \(\mathrm{T}=\mathrm{P}^{\mathrm{a}} \mathrm{d}^{\mathrm{b}} \mathrm{E}^{\mathrm{c}}\)

Taking dimensions of each physical quantity.

\([\mathrm{T}]=\left[\mathrm{ML}^{-1}\mathrm{T}^{-2}\right]^{\mathrm{a}}\left[\mathrm{ML}^{-3}\right]^{\mathrm{b}}\left[\mathrm{ML}^{2} \mathrm{T}^{-2}\right]^{\mathrm{c}}\)

Equating the exponents of \(\mathrm{M}, \mathrm{L}\) and \(\mathrm{T}\) on both the sides,

\(\mathrm{M}^{\mathrm{a}+\mathrm{b}+\mathrm{T}} \mathrm{L}^{-\mathrm{a}-3 \mathrm{b}+2 \mathrm{c}} \mathrm{T}^{-2 \mathrm{a}-2 \mathrm{c}} =\mathrm{T}\)

\(\mathrm{a}+\mathrm{b}+\mathrm{c} =0\)

\(-\mathrm{a}-3 \mathrm{b}+2 \mathrm{c} =0\)

\(-2 \mathrm{a}-2 \mathrm{c} =1\)

Solving these equations for \(a\), \(b\) and \(c,\) we get

\(a=-\frac{5}{6}, b=\frac{1}{2}\) and \(\frac{1}{3}\)