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

Options

A. a=b
B. `a+b=n^(2)`
C. `a=nb`
D. `a+b=2^(n//2)`

Answer

To solve the problem, we need to find the relationship between the coefficients \( a \) and \( b \) of \( x^r \) and \( x^{n-r} \) in the expansion of \( (1 + x)^n \).

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term in the binomial expansion of \( (1 + x)^n \) is given by:
\[ T_{k+1} = \binom{n}{k} x^k \]
where \( k \) varies from \( 0 \) to \( n \).

2. **Find the Coefficient of \( x^r \)**:
The coefficient of \( x^r \) in the expansion is:
\[ a = \binom{n}{r} \]

3. **Find the Coefficient of \( x^{n-r} \)**:
Similarly, the coefficient of \( x^{n-r} \) in the expansion is:
\[ b = \binom{n}{n-r} \]

4. **Use the Property of Binomial Coefficients**:
We know from the properties of binomial coefficients that:
\[ \binom{n}{n-r} = \binom{n}{r} \]
This means that:
\[ b = \binom{n}{n-r} = \binom{n}{r} = a \]

5. **Conclusion**:
Therefore, we conclude that:
\[ a = b \]

### Final Answer:
The relationship between the coefficients \( a \) and \( b \) is:
\[ a = b \]