How many possible different weekly schedules are there if an employee works five full days and two half-days? A firm works $7$ days a week.
Every employee must work exactly $5$ full days and $2$ half-days each week.
A half-day can be either morning or afternoon, and two half-days cannot be held on the same day.
How many possible different weekly schedules are there? 
I have tried $5{7\choose5}+2{7\choose2}$ but i am still getting the incorrect answer.
The correct answer is $84$.
Can someone explain what I am doing wrong in my working out?
Thanks
 A: *

*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 $4 \cdot \binom{7}{5}=84.$
A: 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.
A: First we will select 5 Full days in 7C5 = 21 ways.
Now on to half days each day can be worked in 2 ways =2x2 = 4
Hence for a week it will be 21x4=84 ways.
