Which type of carrier is required by law to convey passengers or freight without refusal if the approved fare or charge is paid?

The correct choice emphasizes the nature of a common carrier, which is a business that provides transportation services to the general public under a regulatory framework. Common carriers are legally obligated to transport anyone who pays the agreed fare, barring any legal exceptions, such as safety or capacity constraints. This requirement ensures that they serve the public and maintain an open door policy for potential customers.

In contrast, private carriers, which are often used for specific individuals or companies, have the discretion to refuse service since they are not offering transportation to the general public. Similarly, contract carriers operate under specific agreements and may choose whom to serve based on the terms of their contracts. Special carriers, while not as commonly defined, refer typically to transport providers that may have specific parameters guiding their operations and service refusals. Therefore, the distinction lies in the common carrier's legal obligation to serve all paying customers as a key component of its business model.