In the expansion of `(2x+y)^(3)-(2x-y)^(3)` , then coefficient of `x^(2)` y is :

Options

A. 24
B. 18
C. 12
D. 16

Answer

To solve the problem of finding the coefficient of \( x^2 y \) in the expression \( (2x + y)^3 - (2x - y)^3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Expression**:
We need to simplify the expression \( (2x + y)^3 - (2x - y)^3 \).

2. **Apply the Binomial Expansion**:
We will use the binomial expansion formula:
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
and
\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]
Here, let \( a = 2x \) and \( b = y \).

3. **Expand \( (2x + y)^3 \)**:
\[ (2x + y)^3 = (2x)^3 + 3(2x)^2(y) + 3(2x)(y^2) + y^3 \]
\[ = 8x^3 + 12x^2y + 12xy^2 + y^3 \]

4. **Expand \( (2x - y)^3 \)**:
\[ (2x - y)^3 = (2x)^3 - 3(2x)^2(y) + 3(2x)(y^2) - y^3 \]
\[ = 8x^3 - 12x^2y + 12xy^2 - y^3 \]

5. **Subtract the Two Expansions**:
Now, we subtract the second expansion from the first:
\[ (2x + y)^3 - (2x - y)^3 = (8x^3 + 12x^2y + 12xy^2 + y^3) - (8x^3 - 12x^2y + 12xy^2 - y^3) \]

6. **Combine Like Terms**:
Simplifying the expression:
\[ = 8x^3 + 12x^2y + 12xy^2 + y^3 - 8x^3 + 12x^2y - 12xy^2 + y^3 \]
\[ = 0 + 24x^2y + 2y^3 \]

7. **Identify the Coefficient of \( x^2y \)**:
From the simplified expression \( 24x^2y + 2y^3 \), we see that the coefficient of \( x^2y \) is \( 24 \).

### Final Answer:
The coefficient of \( x^2 y \) is \( 24 \).