If a and b are respective coefficients of `x^m` and `x^n` in the expansion of `(1+ x)^(m+n)` then

Options

A. a + b = m+ n
B. a = 2b
C. a = b
D. b = 2a

Answer

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

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The binomial theorem states that:
\[ (1 + x)^{k} = \sum_{r=0}^{k} \binom{k}{r} x^r \]
where \( \binom{k}{r} \) is the binomial coefficient, which gives the coefficient of \( x^r \) in the expansion.

2. **Finding Coefficient \( a \)**:
To find the coefficient \( a \) of \( x^m \) in the expansion of \( (1 + x)^{m+n} \):
\[ a = \binom{m+n}{m} \]
This is because the coefficient of \( x^m \) corresponds to choosing \( m \) terms from \( m+n \).

3. **Finding Coefficient \( b \)**:
Similarly, to find the coefficient \( b \) of \( x^n \) in the same expansion:
\[ b = \binom{m+n}{n} \]
This is because the coefficient of \( x^n \) corresponds to choosing \( n \) terms from \( m+n \).

4. **Using the Property of Binomial Coefficients**:
We know from the property of binomial coefficients that:
\[ \binom{m+n}{m} = \binom{m+n}{n} \]
This means that \( a = b \).

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

### Final Answer:
The coefficients \( a \) and \( b \) are equal, i.e., \( a = b \).

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