Final answer:
Using the Pigeonhole Principle, 31 random samples are needed to ensure that at least 7 samples of one type of ice cream have been eaten, among 5 different types. This is because after 30 samples (6 of each type), the 31st sample will push one type to at least 7.
Step-by-step explanation:
To determine how many random samples of ice cream must be eaten to guarantee that at least 7 samples of one type have been eaten, given that there are 5 different types of ice cream, we can use the Pigeonhole Principle.
The Pigeonhole Principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon. In this case, the 'pigeons' are the samples of ice cream, and the 'pigeonholes' are the different types of ice cream.
If we have 6 samples of each type of ice cream, we will have a total of 5 types x 6 samples = 30 samples, and we can't guarantee that any type has been sampled 7 times.
However, once we add one more sample (the 31st sample), we are guaranteed that at least one type of ice cream has been sampled at least 7 times. Therefore, the answer is 31, which corresponds to option (a).