Derive the relationship between half life and decay constant of a radioelement.

How many $\mathrm{\alpha }$ and ${\mathrm{\beta }}^{-}$ particles are emitted in the transmutation ${}_{90}{}^{232}\mathrm{Th}⟶{}_{82}{}^{208}\mathrm{Pb}?$

Find the number of $\mathrm{\alpha }$ and ${\mathrm{\beta }}^{-}$ particles emitted in the process ${}_{86}{}^{222}\mathrm{Rn}\to$

Note: In the textbook the product of the reaction is not present, the correct question will be as follows 

Find the number of $\mathrm{\alpha }$ and ${\mathrm{\beta }}^{-}$ particles emitted in the process ${}_{86}{}^{222}\mathrm{Rn}\to {}_{82}{}^{206}\mathrm{Pb}$?

How many $\mathrm{\alpha }$-particles are emitted by $0.1 \mathrm{g}$ of ${}_{}{}^{226}\mathrm{Ra}$ in one year?