How many possible different weekly schedules are there if an employee works five full days and two half-days?

• no. of ways to choose the $5$ full days is $\binom{7}{5}$.
• Once those are chosen, the two half days are also simultaneously chosen. So you cannot make any independent choices for them anymore. Think of it as follows: if we choose M, Tu, We, Fr, Su as full days then automatically Thu, Sat will have to be half days.
• Now on those two half-days, one can either work in the morning or in the evening. So there are $4$ ways to do this $MM, ME, EM, EE$.

So the total no. of ways is $4 \cdot \binom{7}{5}=84.$

The error you have made is in choosing the days to work half days and choosing the days to work full days separately.

Out of the $7$ days, the employee must work every day. Two of those will be half days. This gives us an answer of $$2^2\cdot\binom{7}{2} = 84$$

The employee must choose which two days to work half days, and whether to work in the morning or afternoon.