why isn't the counting principle giving the right answer? Note : This is not homework, it is self-study. 

An employer interviews eight people for four openings in the company. Three of the eight people are women. If all eight are qualified, in how many ways could the employer fill the four positions if 
  a)the selection is random
  and
  b)exactly two are women?

Part a) is already taken care of, now for part b). My reasoning goes like this: 
There must be $2$ women chosen, so imagine that for the first position we choose a woman. There's $3$ ways to choose. For the second position we choose another woman, there's $2$ ways to choose. For the third position we choose a man, there's $5$ ways to choose. For the fourth position we choose another man, there's $4$ ways to choose. Then we multiply: $3\cdot2\cdot5\cdot4=120$. 
But the answer in the book is $30$. 
 A: You can't simply multiply like that to get the correct number of combinations. What you are doing gives you the number of permutations, but as in this case order does not matter, that is incorrect. 
Consider selecting $2$ women from a group of $3$ women. Your logic suggests there are six different possible combinations, which is clearly incorrect as there are $3$ combinations (as order does not matter). There are however, six permutations. 
Once you've established the difference between the two and notice you require a combination for this question, your result is 
given by 
$$
{3 \choose 2} \cdot {5 \choose 2} = 30
$$
A: The book’s answer assumes that the positions are indistinguishable, i.e., that the question is simply asking how many ways there are to choose $2$ women and $2$ men; Donkey Kong’s answer explains the reasoning behind the resulting calculation.
Your calculation assumes that the positions are distinguishable: that putting persons $A,B,C$, and $D$ in positions $1,2,3$, and $4$, respectively, is different from putting person $A$ in position $2$, person $B$ in position $1$, and persons $C$ and $D$ in positions $3$ and $4$, respectively. That is also how I would read the question. However, in that case your answer is still not right, though it’s closer than the one in the book. The problem is that your calculation doesn’t account for the assignments in which the women do not occupy the first two positions. To get the correct result using your interpretation, start with the calculation of $30$ ways to choose $2$ men and $2$ women. Then note that you can assign the $4$ chosen applicants to the $4$ positions in $4!$ different ways, for a total of $30\cdot4!=720$ different ways to fill the positions.
A: We need to choose $2$ women from the available $3$ and then choose $2$ men from the available $5$. This gives us $${3 \choose 2}\cdot {5 \choose 2} = 3 \cdot 10 = 30.$$
