If m and n are positive integers, then prove that the coefficients of `x^(m) " and " x^(n)` are equal in the expansion of `(1+x)^(m+n)`

Explanation

To prove that the coefficients of \( x^m \) and \( x^n \) are equal in the expansion of \( (1+x)^{m+n} \), we will use the Binomial Theorem, which states that:

\[ (1+x)^{k} = \sum_{r=0}^{k} \binom{k}{r} x^r \]

where \( \binom{k}{r} \) is the binomial coefficient representing the coefficient of \( x^r \) in the expansion.

### Step-by-Step Solution:

1. **Identify the Expansion**:
We are considering the expansion of \( (1+x)^{m+n} \).

2. **Write the General Term**:
The general term in the expansion of \( (1+x)^{m+n} \) is given by:
\[ T_r = \binom{m+n}{r} x^r \]
where \( r \) varies from \( 0 \) to \( m+n \).

3. **Find the Coefficient of \( x^m \)**:
The coefficient of \( x^m \) in the expansion is:
\[ \text{Coefficient of } x^m = \binom{m+n}{m} \]

4. **Find the Coefficient of \( x^n \)**:
Similarly, the coefficient of \( x^n \) in the expansion is:
\[ \text{Coefficient of } x^n = \binom{m+n}{n} \]

5. **Use the Property of Binomial Coefficients**:
We know from the properties of binomial coefficients that:
\[ \binom{m+n}{m} = \binom{m+n}{n} \]
This is because choosing \( m \) objects from \( m+n \) is equivalent to choosing \( n \) objects from \( m+n \) (since \( n = (m+n) - m \)).

6. **Conclusion**:
Since the coefficients of \( x^m \) and \( x^n \) are equal, we have:
\[ \binom{m+n}{m} = \binom{m+n}{n} \]
Therefore, we have proved that the coefficients of \( x^m \) and \( x^n \) in the expansion of \( (1+x)^{m+n} \) are equal.