Answer
${\bf 2.73 \times 10^{17}}\;\rm particle$
Work Step by Step
We need to determine the total number of beta particles emitted after a sample of $ ^{137}\text{Cs} $ (Cesium-137) has fully decayed.
So we need to calculate the total number of $ ^{137}\text{Cs} $ atoms originally present, which will tell us the total number of beta particles emitted by the time the sample has fully decayed.
The decay constant $ r $ is given by
$$
r = \frac{\ln(2)}{t_{_{1/2}}}
$$
where the half-life $ t_{1/2} $ of $ ^{137}\text{Cs} $ is 30 years
$$
r = \frac{\ln(2)}{ 30\times 365\times 24\times 3600}
$$
$$
r =\bf 7.33 \times 10^{-10} \, \rm {s}^{-1}
$$
Now we need to calculate the Initial Number of Atoms $ N_0 $.
Recalling that
$$
R_0 = r N_0
$$
where the initial activity $ R_0 $ is related to the decay rate $ r $ and the initial number of atoms $ N_0 $ by this formula.
Solve for $ N_0 $:
$$
N_0 = \frac{R_0}{r}
$$
where the initial activity $ R_0 $ of the cesium sample is $ 2.0 \times 10^8 \, \text{Bq} $
$$
N_0 = \frac{2.0 \times 10^8 }{7.33 \times 10^{-10} }=\bf 2.73 \times 10^{17} \, \rm {atom}
$$
Since each $ ^{137}\text{Cs} $ atom decays by emitting one beta particle, the total number of beta particles emitted over time (when the sample has fully decayed) will equal the initial number of atoms.
Thus,
$$
N_{\alpha} = N_0 = \color{red}{\bf 2.73 \times 10^{17}}\;\rm particle
$$
