Start by finding the prime factorization of 378:
378 = 2(189)
= 2(9)(21)
= 2(3)(3)(3)(7)
= (2^1)(3^3)(7^1)

The fastest way to find the number of factors of a large number is to take the exponent of each of the prime factors, raise it by one, and multiply them all together. In this case, the exponents are 1, 3, and 1. Raise them each by one, and you have 2, 4, and 2. Multiply them all together, and the result is 2(4)(2) = 16, choice (C).