Question 50

1. The hash table contains $100$ slots: $\{0, 1, 2, \dots, 99\}$.

2. Under the simple uniform hashing assumption:

   • Each key is equally likely to hash to any of the $100$ slots, independently of where other keys have hashed.
   • For any single insertion, the probability of choosing any specific slot is $\frac{1}{100}$.

3. Condition for the First Three Slots Remaining Unfilled:

   • The first three slots (slots $0, 1, 2$) must receive zero keys.
   • That is, each inserted key must hash into one of the remaining $100 - 3 = 97$ slots.
   • For any individual key, the probability that it falls into one of the other $97$ slots is:
     $$P = \frac{97}{100}.$$

4. Three Independent Insertions:

   • Because the three key insertions are independent events:
     $$P(\text{all 3 keys avoid first 3 slots}) = \left(\frac{97}{100}\right)^3 = \frac{97 \times 97 \times 97}{100^3}.$$

This corresponds to option A.