Final answer:
There are 129 ways to split a class of 130 students into two non-empty groups when the order of the groups is not important. This is found by considering the number of ways to choose a subset for the first group and avoiding double-counting since the order does not matter.
Step-by-step explanation:
To find the number of ways that a class of 130 students can be split into two non-empty groups where the order of the groups is not important, one can think of it as a problem of combinatorics. When splitting a group into two, you can think of picking a subset for the first group; the rest of the students make up the second group. Since the order does not matter (Group 1 and Group 2 are identical to Group 2 and Group 1), we should avoid double-counting by dividing the total combinations by 2.
You can split the first group with anywhere from 1 to 129 students (you cannot have 0 or all 130 because the groups are non-empty and distinct). For each 'n' number of students in the first group, there is exactly one corresponding configuration of the second group with the remaining students.
Therefore, the number of ways to split the class is the same as the number of ways to choose a group without considering the order: 129 ways. This corresponds to option (a), which is the correct answer to the given multiple-choice question.