A board game has a round that starts with three tosses of a fair coin, followed by one throw of a…

Step 1. The sample space. The three tosses give $2^3 = 8$ strings. The die adds a factor $6$. So $n(S) = 8 \times 6 = 48$.

Step 2. Exactly two heads. The strings are $HHT$, $HTH$ and $THH$. That is $3$ strings.

Step 3. An even number. The die shows $2$, $4$ or $6$. That is $3$ results.

Step 4. Together. Each of the $3$ strings goes with each of the $3$ results, so there are $3 \times 3 = 9$ outcomes.