Counting and Probability. Computer Science Majors and Positions. I have found myself overwhelmingly frustrated with "counting." 
Please help with the following problem:
There are $30$ students in my class, and $8$ are computer science majors.
I randomly choose five (different) students to serve as officers in the class: President,
Vice-President, Secretary, Treasurer, and Omsbudsman. 
a) What is the probability that exactly two of the five officers selected are computer
science majors?
b) What is the probability that the President is a computer science major given that
exactly two officers are computer science majors?
I was actually able to solve a). I did the following Let $A$ be defined as "exactly two of the five officers selected are computer
science majors", then
$P(A)=\frac{{8\choose 2}{22\choose 3}}{{30 \choose 5}}$. To be honest though I wasn't even entirely convinced that I was on the right track until I checked my answer. 
Now, for b) I am at a complete loss. I am really frustrated and I feel like not being able to count in these ways is holding me back immensely. Is there anyway that someone can help me with this problem and also please refer me to a free online even an offline source where I can find many similar counting practice examples or maybe a methodic way to train myself on how to count. Everytime I try to google I end up with elementary methods of counting. It seems to me that this comes very naturally to my professor and when he is explaining his steps outloud I am convinced. However, almost everytime I try to approach a counting problem by myself am lost. 
Thank you
 A: (a) is correct.


*

*The favoured space are ways to select two computer majors and thee non-computer majors in the other position.

*The total space are ways to select any five students for the position.
$$P(A) = \frac{\binom{8}{2}\binom{22}{3}}{\binom{30}{5}}$$
For (b) When given 2 computer majors and 3 other students, what is the probability that the one who was selected for president was one of the computer majors?

Alternatively:


*

*The favoured space is ways to select 2 computers majors, 3 other students, and select one of the two computer majors for president.

*The total (conditioned space) is ways to select 2 computers majors, 3 other students, then select any one of these five for president.


*

*Cancel the obvious common terms.


A: "What is the probability that the President is a computer science major given that exactly two officers are computer science majors?"  You have 5 officers and you say exactly 2 are CS majors.  That's enough information to answer the question.  You can ignore all the stuff leading up to that.  What is the probability that the a particular officer is one of the 2 CS major officers when there are 5 officers?  The probability that the president is a CS major is is 2/5.
