I'm having trouble understanding when to use combinations for counting, and when to multiply probabilities. Say the question is what's the probability of drawing $2$ diamonds and one card that isn't a diamond.
Why is one of these approaches wrong:
$$\dfrac{_{13}C_2\,\cdot\, _{39}C_1}{_{52}C_3}$$ Here, I am choosing $2$ diamonds from the $13$ diamonds, times choosing $1$ non-diamond from the $39$ non-diamonds, all over the total number of ways you can choose $3$ cards.
$$\frac{13}{52}\cdot\frac{12}{51}\cdot\frac{39}{50}$$ Here, first the probability of choosing $1$ diamond, then the second diamond from the remaining $51$ cards and $12$ diamonds, and then finally choosing one non-diamond from the remaining $50$ cards.
These both give $2$ different answers. They're off by a factor of $3$. Which is correct? Why?
