Question

question 2 of 10
which of the following statements regarding the expansion of ((x + y)^n) are correct?

a. the coefficients of (x^{n - 1}) and (y^{n - 1}) both equal 1.

b. for any term (x^a y^b) in the expansion, (a - b = n).

c. for any term (x^a y^b) in the expansion, (a + b = n)

d. the coefficients of (x^n) and (y^n) both equal 1.

Explanation:

Step1: Recall binomial expansion formula

The expansion of $(x+y)^n$ is $\sum_{k=0}^n \binom{n}{k}x^{n-k}y^k$, where each term is $\binom{n}{k}x^{a}y^{b}$ with $a=n-k$, $b=k$.

Step2: Check Option A

Coefficient of $x^{n-1}$: $\binom{n}{1}=n$; coefficient of $y^{n-1}$: $\binom{n}{n-1}=n$. Not 1. A is wrong.

Step3: Check Option B

For any term, $a+b=(n-k)+k=n$, not $a-b=n$. B is wrong.

Step4: Check Option C

From Step1, $a+b=n$. C is correct.

Step5: Check Option D

Coefficient of $x^n$: $\binom{n}{0}=1$; coefficient of $y^n$: $\binom{n}{n}=1$. D is correct.

Answer:

C. For any term $x^a y^b$ in the expansion, $a + b = n$, D. The coefficients of $x^n$ and $y^n$ both equal 1.