If A and B are the coefficients of `x^n` in the expansion `(1 + x)^(2n)` and `(1 + x)^(2n-1)` respectively, then

Options

A. A= B
B. 2A = B
C. A = 2B
D. none of these

Explanation

To solve the problem, we need to find the coefficients \( A \) and \( B \) in the expansions of \( (1 + x)^{2n} \) and \( (1 + x)^{2n-1} \) respectively, and then establish a relationship between them.

### Step-by-step Solution:

1. **Identify the Coefficient \( A \)**:
The coefficient \( A \) of \( x^n \) in the expansion of \( (1 + x)^{2n} \) can be obtained using the binomial theorem:
\[ A = \binom{2n}{n} \]

2. **Rewrite \( A \)**:
We can express \( A \) using the identity for binomial coefficients:
\[ A = \binom{2n}{n} = \frac{2n!}{n!n!} \]
We can also relate this to another binomial coefficient:
\[ A = \frac{2n}{n} \cdot \binom{2n-1}{n-1} = 2 \cdot \binom{2n-1}{n-1} \]

3. **Identify the Coefficient \( B \)**:
The coefficient \( B \) of \( x^n \) in the expansion of \( (1 + x)^{2n-1} \) is:
\[ B = \binom{2n-1}{n} \]

4. **Rewrite \( B \)**:
We can also express \( B \) using the identity for binomial coefficients:
\[ B = \binom{2n-1}{n} = \binom{2n-1}{n-1} \]

5. **Establish the Relationship**:
Now, we have:
\[ A = 2 \cdot \binom{2n-1}{n-1} \]
and
\[ B = \binom{2n-1}{n-1} \]
Therefore, we can relate \( A \) and \( B \):
\[ A = 2B \]

### Conclusion:
Thus, the relationship between the coefficients \( A \) and \( B \) is:
\[ A = 2B \]