The set S is defined as {1, -2, -3, 8, 21, 30, -22, -5, 6, -

The wording doesn't make sense, because they're saying chi is "defined" in a certain way, and chi is not well-defined. What is the chi of this set? Who knows, there are dozens of possible values.

Anyway, if we need a value greater than 7, we certainly need the value to be positive. The set contains 5 positive and 5 negative numbers. If we'll pick three of them, and the product will be positive, we must either pick 3 positive numbers, or 1 positive and 2 negative numbers. We can pick 3 positive numbers in 5C3 = (5)(4)/2! = 10 ways. We can pick one positive and two negatives in 5C1*5C2 ways, or (5)*(5)(4)(3)/3! = 50 ways. So in total, there are 10+50 = 60 ways to get a positive product. We then need to rule out any positive products that will be less than 7, but we only get a positive product less than 7 if we choose specifically the values 1, -2 and -3, so in only one way, leaving 59 positive products greater than 7. There is one last thing to check - that you can't get equal values of the product in two different ways - but glancing at the set, that doesn't appear to be possible.