To facilitate development of telephone service in a rural province

The corporation's argument is that the provincial government would "profit greatly" because "all calls would be routed through the province," leading to a large increase in the subsidy received.

Let's break down why option B is the crucial assumption:

The core of the corporation's argument: The corporation says the provincial government will profit because all calls would be routed through the province, and the provincial government gets a subsidy for long-distance calls "going into the province."
The problem: If the subsidy only applies to calls terminating in the province, and the corporation's plan routes all long-distance calls (even those just passing through) through the province, then the provincial government would only get a subsidy for the calls actually going into the province, not for the massive increase in calls just passing through.
The necessary assumption: For the provincial government to "profit greatly" from all calls being routed through the province, the national government's subsidy must apply not just to calls terminating in the province, but also to at least some long-distance calls that are merely routed through the province. Without this, the increased call volume due to routing wouldn't translate into increased subsidy for the provincial government.
Let's look at why the other options are not the primary assumption:

A. Without the plan, all long-distance telephone service in the province would involve at least some charges to callers. While this might be true and makes the "no charge" aspect attractive, it doesn't directly relate to how the provincial government would profit from the subsidy based on routing.
C. The provincial government would be interested in splitting its subsidy with the corporation only if doing so would yield significant profits for the province. This is a reasonable expectation for the provincial government, but it's an assumption about the provincial government's motivation, not about how the corporation's plan would actually generate the predicted profit from the subsidy mechanism. The corporation's argument is that it will be profitable.
D. The national government's subsidy for any long-distance call into the province is calculated as a fixed percentage of the charge to the caller. The problem states the corporation would offer calls "without any charge to the callers." If the subsidy is a fixed percentage of the charge, and the charge is zero, then the subsidy would also be zero. This would undermine the corporation's argument, making it an assumption that would work against their claim of profit, not for it.
E. In order for the arrangement to be profitable for the province, the province must receive more from the increased subsidy than it pays the corporation. This is a basic economic principle of profitability, but it's a general truth about profit, not the specific assumption underlying the corporation's claim that the routing of calls would lead to significant profits from the subsidy. The corporation's argument is about how the revenue (subsidy) would increase.
The final answer is
B
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