Let be the random variable for the number of bad oranges picked out. Then
if and zero otherwise, where
is the so-called binomial coefficient.
(a) Of the 6 good oranges, you pick 0. Of the 3 bad oranges, you pick 3. Of the 10 total oranges, you pick 3. So the probability of picking out all bad oranges is
(b) By similar reasoning, the probability of drawing exactly 1 bad orange is
(c) "At least 1 bad orange" means you pick out 1, 2, or 3 bad oranges. These events are mutually exclusive, and we already know the probabilities of picking out exactly 1 or all 3 bad oranges. The remaining probability of drawing 2 bad oranges is
so the overall probability of drawing at least 1 bad orange is
(d) I assume you mean "at most 2 bad oranges," meaning you pick out 0, 1, or 2 bad oranges. Again, these events are mutually exclusive, and the probability of picking out no bad oranges is
hence the probability of drawings at most 2 bad oranges is