Understanding Inclusion-Exclusion principle 
Problem: You have $20$ employees. $4$ of them are women. You have $50$ different jobs to give for your employees, but each women should get at least one job.  

The Proof: 
The author split the problem into cases. 
The first one is,
we'll choose $4$ jobs dedicated for the women, and then we'll assign the rest of the jobs:  
$${50 \choose 4}{46 \choose 20}$$
Next case is where one of the women doesn't get any job:  
$${4 \choose 1}{50 \choose 3}{47 \choose 19}$$
And so on (similar to the above)
Question:
My impression is that the solution is wrong because it doesn't assign all the $50$ jobs! (And that's weird because the guy who solved it got an A on this question) 
Am I right? If so, What should be done differently? 
 A: in this case if its necessary to give minimum one job to any employee but  each women should get at least one job then you must choose 4 job for womans and 16 job for others (surely mans :) ) so you have :
$${50 \choose 4}$$
then you have 46 job that can be given to 20 employee that are able to have more than one job then you have this :
$$X_1+X_2+X_3+...+X_{20} =46  , 0<= X_i<=46$$  then we have :
$${46+20-1 \choose 46}$$
finally the answer is : 
$${50 \choose 4}{65 \choose 46}$$
A: You are right - it doesn't assign all the jobs. I am not certain what does it actually count, but surely it's far from the right answer.
Every job needs a person who does it. We can provide a function $f : \{1, 2, ... 50\} \rightarrow \{1,2, ... 20\}$ which assigns a job number, to the person who does the job (i assume that one person can have multiple jobs but one job can be assigned to just one person, which is also not obvious). To count the possible assignments now, we use the Inclusion-Exclusion principle.
The number of all possible discussed functions is $20^{50}$. Then we remove all the job-assigns that do not consider women number $1$. There are $19^{50}$ such functions. Because there are $4$ women, we get $20^{50} - 4*19^{50}$. Next, we must add all the functions we counted multiple times (those, which do not assign any job to 2 women), take again and add again as in the Inclusion-Exclusion principle. The final answer is:
$20^{50} - 4*19^{50} + \binom{4}{2} 18^{50} - \binom{4}{3} 17^{50} + \binom{4}{4} 16^{50} $
which can be shortened to 
$\sum_{i=0..4} (-1)^{i}\binom{4}{i}*(20-i)^{50}$
EDIT (this also answers the comments below the question - I cannot comment yet):
You are asking why your mate's approach is wrong. You see, the Inclusion-Exclusion principle is not exactly counting separately some cases - it's more of an algorithm we use to easly determine the overall number of some, for example, different events (as different assigning of jobs to some people) in the easiest way, and then just checking if we counted too many or too less and substracting or adding the difference respectively. My solution uses the negative approach: we count all the events and just substract the bad ones, because in this particular case it is easier.
Your mate probably tried to do that (the second case which was actually meant to be the second step of the algorithm was to be substracted from the first - am I right?), but he failed and his answer is just wrong: he counted the number of ways to choose jobs done surely by women (but without noticing which one does which job) and then choosing 20 jobs from the jobs left, so that other people are going to do them (again, it doesn't even check the order - its just choosing the jobs to be done). Any other substraction/adding of anything is just making everything worse, so I won't explain the second step.
