 Solution

For each ball, the event of landing in $S_j$ is a Bernoulli distribution trial. Under the null hypothesis its probability is $\epsilon$. Under the selected alternative $Q_j$, its probability is $p_1=\pi +(1-\pi )\epsilon =\epsilon +\pi (1-\epsilon )$. The placements are independent random variables conditional on that selected index, so the count has the binomial distribution:The second assertion concerns $Q_j$, rather than the unconditional mixture model $P_1$.