A farmer has harvested a very large amount of oranges and grapefruits together. Exactly half of them are oranges, and the other half are grapefruits. He wants to take exactly half of the harvest to keep for himself so that he can grow new trees. He cannot distinguish oranges and grapefruits from the outside, and he may not cut them open, so he must choose his half randomly. The farmer does not care if he grows oranges or grapefruits, but his family likes grapefruits so he wants to make sure that he grows at least 1 grapefruit tree with a good probability. (a) Given that the farmer has harvested n oranges and n grapefruits, what is the probability that the farmer picks all oranges? (b) The farmer wants to reassure his family by telling them how low the probability of selecting all oranges is. However, he does not have the time to compute the percentage for part a. Given that he has 1000 oranges and 1000 grapefruits, what is an approximate upper bound for this probability that he can compute easily? (c) Prove that the deathbed formula/bound you used for part (b) is true. Hint: Think about n k as a product of several factors.