If `(1+2x+3x^(2))^(10)=a_(0)+a_(1)x+a_(2)x^(2)+……+a_(20)x^(20)`, then `a_(1)` equals :

Options

A. 10
B. 20
C. 210
D. None of these

Answer

To find the coefficient \( a_1 \) in the expansion of \( (1 + 2x + 3x^2)^{10} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Function**: We start with the function \( f(x) = (1 + 2x + 3x^2)^{10} \).

2. **Differentiate the Function**: To find the coefficient of \( x \) (which is \( a_1 \)), we differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}[(1 + 2x + 3x^2)^{10}] \]

3. **Apply the Chain Rule**: Using the chain rule, we have:
\[ f'(x) = 10(1 + 2x + 3x^2)^9 \cdot (2 + 6x) \]

4. **Evaluate at \( x = 0 \)**: To find \( a_1 \), we substitute \( x = 0 \) into \( f'(x) \):
\[ f'(0) = 10(1 + 2(0) + 3(0)^2)^9 \cdot (2 + 6(0)) \]
Simplifying this gives:
\[ f'(0) = 10(1)^9 \cdot (2) = 10 \cdot 2 = 20 \]

5. **Conclusion**: Thus, the coefficient \( a_1 \) is:
\[ a_1 = 20 \]

### Final Answer:
\[ \boxed{20} \]