Final answer:
A section 501(c)(3) organization is permitted to engage in advancing religious beliefs that differ from mainstream scientific theories. They cannot campaign for political candidates, distribute assets to shareholders, or operate a racially discriminatory private school.
Step-by-step explanation:
Among the activities listed, a section 501(c)(3) organization is permitted to engage in advancing religious beliefs that differ from mainstream scientific theories. Section 501(c)(3) organizations include non-profit organizations like churches and community service groups. Such organizations are allowed to promote their religious beliefs, as long as their activities do not violate public policy or involve political campaign activities.
Engaging in a campaign for a political candidate, distributing assets to shareholders upon dissolution, or operating as a private school that limits admission based on race are all activities that are either restricted or directly prohibited for 501(c)(3) organizations. The distribution of profits to shareholders would violate the requirement that any profits be reinvested back into the organization, and engaging in racial discrimination violates the Civil Rights Act of 1964. Moreover, campaigning for a political candidate can result in the loss of tax-exempt status, according to the IRS regulations prohibiting such involvement in political campaigns.