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

Options

A. A+B
B. A+B=0
C. A = Rb
D. A = nB

Answer

To solve the problem, we need to find the coefficients \( A \) and \( B \) of \( x^r \) and \( x^{n-r} \) respectively in the expansion of \( (1 + x)^n \).

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The binomial expansion of \( (1 + x)^n \) can be expressed as:
\[ (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \]
where \( \binom{n}{k} \) is the binomial coefficient, which gives the coefficient of \( x^k \).

2. **Finding Coefficient \( A \)**:
To find the coefficient \( A \) of \( x^r \), we look for the term where \( k = r \):
\[ A = \binom{n}{r} \]

3. **Finding Coefficient \( B \)**:
Similarly, to find the coefficient \( B \) of \( x^{n-r} \), we look for the term where \( k = n - r \):
\[ B = \binom{n}{n - r} \]

4. **Using the Property of Binomial Coefficients**:
There is a property of binomial coefficients that states:
\[ \binom{n}{r} = \binom{n}{n - r} \]
This means that the coefficients \( A \) and \( B \) are equal:
\[ A = B \]

5. **Conclusion**:
Therefore, we conclude that:
\[ A = B \]
This verifies that the coefficients of \( x^r \) and \( x^{n-r} \) in the expansion of \( (1 + x)^n \) are equal.