Which of the following statements regarding the expansion of $(x+y)^{n}$ are correct? A. For any term $x^{a} y^{b}$ in the expansion, $a+b=n$. B. The coefficients of $x^{n-1}$ and $y^{n-1}$ both equal 1 . C. The coefficients of $x^{a} y^{b}$ and $x^{b} y^{a}$ are equal. D. For any term $x^{a} y^{b}$ in the expansion, $a-b=n$.

Step 1 ：We are given the expression $(x+y)^{n}$ and asked to determine which of the following statements are correct.

Step 2 ：The expansion of $(x+y)^{n}$ is given by the binomial theorem, which states that $(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}$, where $\binom{n}{k}$ is the binomial coefficient, also known as 'n choose k'.

Step 3 ：Statement A: For any term $x^{a} y^{b}$ in the expansion, $a+b=n$. This is true because in each term of the expansion, the powers of $x$ and $y$ add up to $n$.

Step 4 ：Statement B: The coefficients of $x^{n-1}$ and $y^{n-1}$ both equal 1. This is not necessarily true. The coefficient of $x^{n-1}y$ and $x y^{n-1}$ is $n$, not 1.

Step 5 ：Statement C: The coefficients of $x^{a} y^{b}$ and $x^{b} y^{a}$ are equal. This is true because the binomial coefficient $\binom{n}{k}$ is symmetric, i.e., $\binom{n}{k} = \binom{n}{n-k}$.

Step 6 ：Statement D: For any term $x^{a} y^{b}$ in the expansion, $a-b=n$. This is not true. The difference between the powers of $x$ and $y$ in a term does not necessarily equal $n$.

Step 7 ：From the above analysis, we can conclude that the correct statements regarding the expansion of $(x+y)^{n}$ are A and C.

Step 8 ：Final Answer: \(\boxed{\text{A, C}}\)