odds of picking exactly 2 women and 2 men out of 12 men and 12 womem I understand the answer to be 12 choose 2 * 12 choose 2 over 24 choose 4. 
I don't really understand why, or what principle I can extract from the problem. I can understand that we are putting the total possible outcomes in the denominator, but not how the numerator represents the exact 2 by 2 requirement the problem states

There are ${12}\choose{2}$ ways to pick $2$ women out of $12,$ without regards to order (i.e. Mary & Lucy $=$ Lucy & Mary). For each of these ways, there are ${12}\choose{2}$ ways to choose $2$ men. So you must multiply to get the number of possibilities: ${12}\choose{2}$ $\cdot$ ${12}\choose{2}$. Assuming equal likelihood of any choice, divide this by the total number of outcomes ${24}\choose{4}$ to get the probability of any particular group of $4.$
