Question

Which of the following statements regarding the expansion of (x + y)^n are correct?

Answer 1

Answer:

A, B and D are true statements.

Step-by-step explanation:

We are given a binomial expansion

$(x+y)^n=^nC_0x^ny^0+^nC_1x^{n-1}y^1+^nC_2x^{n-2}y^2+...........+^nC_nx^0y^n$

$(x+y)^n=x^n+nx^{n-1}y+^nC_2x^{n-2}y^2+...........nxy^{n-1}+y^n$

Now we will check each option

Option A: The coefficients of $x^n$ and $y^n$ both equal 1.

If we see first and last term of the expansion, This statement is true.

Option B: For any term $x^ay^b$ in the expansion, a + b = n.

Let we take 3rd term of expansion $^nC_2x^{n-2}y^2$

Here, a=n-2 and b=2

If we do  a+b = n-2+2=n

a+b=n is true statement.

Option C: For any term x^ay^b in the expansion, a - b = n.

Let we take 3rd term of expansion $^nC_2x^{n-2}y^2$

Here, a=n-2 and b=2

If we do  a-b = n-2-2=n-4≠n

a-b=n is false statement.

Option D: The coefficients of x^ay^b and x^by^a are equal.

If we take second term from beginning and last of the expansion.

$\text{Coefficient From beginning } nx^{n-1}y=n$

$\text{From last } nxy^{n-1}=n$

This statement true.