Final answer:
To find the coefficient of the x⁴-term in the binomial expansion of (x³)¹², you can use the binomial theorem. Option c is correct.
Step-by-step explanation:
To find the coefficient of the x⁴-term in the binomial expansion of (x³)¹², we can use the binomial theorem. The binomial theorem states that for a binomial expression raised to a power, the coefficient of a specific term can be found using the formula: C(n, r) * a^(n-r) * b^r, where n is the power, r is the term number, a is the coefficient of the first term, and b is the coefficient of the second term.
In this case, our binomial expression is (x³)¹², so a = x³, b = 1, n = 12, and we want to find the coefficient of the x⁴-term, which corresponds to r = 8.
The question asks for the coefficient of the x⁴-term in the binomial expansion of (x³)¹². When expanding the expression, the power on the x gets multiplied by the power outside the parentheses due to the rules of exponents, as in (xa)b = xa·b. Since we are looking for the x⁴-term, and the given expression is (x³)¹², which simplifies to x¹⁶, there is no x⁴-term in this expansion because x⁴ is not a factor of x¹⁶.
Using the formula, we can calculate: C(12, 8) * (x³)^(12-8) * 1^8 = C(12, 8) * x⁴ * 1 = C(12, 8) * x⁴
The coefficient of the x⁴-term is therefore C(12, 8).