\({ }_{92}^{238} \mathrm{U}\) is known to undergo radioactive decay to form by \({ }_{82}^{206} \mathrm{~Pb}\) emitting …

\({ }_{92}^{238} \mathrm{U}\) is known to undergo radioactive decay to form by \({ }_{82}^{206} \mathrm{~Pb}\) emitting alpha and beta particles. A rock initially contained \(68 \times 10^{-6} \mathrm{~g}\) of \({ }_{92}^{238} \mathrm{U}\). If the number of alpha particles that it would emit during its radioactive decay of \({ }_{92}^{238} \mathrm{U}\) to \({ }_{82}^{206} \mathrm{~Pb}\) in three half-lives is \(\mathrm{Z} \times 10^{18}\), then what is the value of \(\mathrm{Z}\) ?

[JEE Advanced 2020, Paper-1]