Suppose you have N objects and N buckets. For each object, randomly place it into a bucket, with each of the N options being equally likely. Through this random process, some buckets may end up empty, while other buckets may have multiple objects. If N=3, there are 33 = 27 ways the three objects can be placed into the three buckets. For each bucket, see if you can verify that

• There is a probability of 8/27 that this bucket contains zero objects

• There is a probability of 12/27 that this bucket contains exactly one objects

• There is a probability of 6/27 that this bucket contains exactly two objects

• There is a probability of 1/27 that this bucket contains exactly three objects When each of N objects is randomly placed into one of N buckets, let P(N) be the probability that a bucket contains exactly 0, 1, or 2 objects. For example, we can use the calculations above to determine that P(3) = 8/27 + 12/27 + 6/27 = 26/27 = 96.3%.

Determine P(100).