Final answer:
The correct statements regarding the expansion of (x + y)ⁿ are A, C, and D, which state (A) the sum of the exponents equals n, (C) the coefficients of terms with reversed exponents are equal, and (D) the coefficients of xⁿ and yⁿ are both 1.
Step-by-step explanation:
The statement "Which of the following statements regarding the expansion of (x + y)ⁿ are correct?" pertains to the binomial theorem and its properties. Let's evaluate the statements given:
A. For any term xᵃyᵇ in the expansion, a + b = n. This statement is correct. According to the binomial theorem, the sum of the exponents in any term of the expansion equals the power n of the binomial.B. For any term xᵃyᵇ in the expansion, a - b = n. This statement is incorrect. The difference of the exponents doesn't equal the power n; rather, their sum does, as stated in option A.C. The coefficients of xᵃyᵇ and xᵇyᵃ are equal. This statement is correct. By the symmetry of the binomial theorem, these two terms are mirror images of each other about the center of the expansion, thus they have equal coefficients.D. The coefficients of xⁿ and yⁿ both equal 1. This statement is correct. The first and last terms of the binomial expansion are always raised to the power n, with the other variable not present; hence their coefficients are 1.
In summary, the correct statements regarding the expansion of (x + y)ⁿ are A, C, and D.