A committee consists of five Mexicans, two Asians, three African Americans, and two Caucasians. A subcommittee of 4* is chosen at random. What is the probability that all the ethnic groups are represented on the subcommittee?
Note: original posting said 5 but I meant 4. Also, I know, the race thing is weird, but it was in my book.
It seems to me like the answer should be $$\frac{\frac{5*2*3*2}{4!}}{\binom{12}{4}}$$ because we want both the numerator and the denominator to be unordered selection.
Alternatively, we would say that it is $$\frac{5*2*3*2}{12*11*10*9}$$ However, something in my reasoning is incorrect as the solution is just $$\frac{5*2*3*2}{\binom{12}{4}}$$ I am dreadfully confused.
source: http://www.math.illinois.edu/~psdey/stat20SU07/Solutions2_2007.pdf