Forming a Team of Professionals

The project manager can be picked in $10$ ways. The deputy project managers (indistinguishable) can be picked in $\frac{9 \times 8}{2!}$ ways, and by the same reasoning, the engineers can be chosen in $\frac{7 \times 6 \times 5 \times 4 \times 3}{5!}$ ways.

In total, there are $10 \times \frac{9 \times 8}{2!} \times \frac{7 \times 6 \times 5 \times 4 \times 3}{5!} = 7560$ ways to form the team.

Number of ways of picking $8$ people from a group of $10$ is, ${10 \choose 8 }=45$ Number of ways of picking $5$ engineers from the above $8$ people is, ${8 \choose {5}}=56$ Number of ways of picking $2$ deputy project managers from the $3$ people left is, ${3\choose 2}=3$ Number of ways of picking $1$ project manager from the $1$ person left is, ${1\choose 1}=1$ Therefore, the total ways of to form a team meeting the required conditions is, ${10 \choose 8}\cdot {8 \choose 5}\cdot {3\choose 2}\cdot {1\choose 1}=45\cdot 56\cdot 3\cdot 1=\boxed{7560}$