This isn't a necessary assumption. Notice that if we negated it, (A) would only tell us that some of the 70 opera companies were not commercially viable. But some of these companies not being viable wouldn't destroy the argument that public interest has driven the founding of the 45 new opera companies, so we can't say that (A) is a necessary assumption.

This is a necessary assumption, because it tells us that the fact that 45 of the currently active opera companies were founded in the last 30 years actually represents a net increase. Let's think about what it would mean to negate (B): there were at least (i.e., more than or equal to) 45 professional opera companies that were active 30 years ago and ceased operating during the last 30 years. It's really important to see that we don't negate or change anything in the phrase "active 30 years ago and that ceased operations during the last 30 years" because that is a descriptor of the kind of company we're talking about: companies that existed 30 years ago and then shut down in the last three decades, so were removed from the equation.

If there were exactly 45 such companies, then the 45 new companies founded in the last 30 years that have lasted until today actually don't represent a net increase in the number of active opera companies: they perfectly balance out the number of opera companies that closed in the last three decades. If there were more than 45 such companies, then even though 45 new companies were founded and have lasted until today, the total number of opera companies has gone down over the last 30 years. This would destroy the argument, which points to the assumed increase in the number of opera companies as evidence for an increase in people's enjoyment of opera.

Irrelevant. We're not interested in other performing arts. The stimulus only talks about opera.

This isn't a necessary assumption. Even if this weren't true — if the average audience size had stayed the same, or even decreased — this could be offset by the apparent increase in the number of opera companies, which might suggest that the total number of people viewing opera could still have increased. Audiences might not have increased in size, but there could still be more audiences overall. So negating (D) wouldn't destroy the argument.

This is a tricky answer choice, because it does point to one of the assumptions we identified: that the primary cause behind the founding of these new companies was audience enthusiasm. But it's crucial to see that this assumption does not require every single one of the 45 companies to have been founded because of audience enthusiasm. If 43 companies were founded because of audience enthusiasm, and 2 were founded just because an older company split, then it would still be fair to say that the increase in opera companies suggests an increase in audience enthusiasm.

So notice that negating this answer choice wouldn't actually destroy the argument: "the 45 most recently founded opera companies were not all established as a result of enthusiasm on the part of a potential audience." The key here is knowing how to negate "all." This statement would destroy the argument if it said that "none" of these 45 companies were founded because of audience enthusiasm. But "all" simply negates to "not all" or "some... not": some of these opera companies were not founded because of audience enthusiasm. As we saw above, this wouldn't destroy the argument, so we can't say that (E) is a necessary assumption.