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 \(x^n\) in the expansion of \((1+x)^{m+n}\) are equal, we will use the binomial theorem.

### Step-by-Step Solution:

1. **Recall the Binomial Theorem**:
The binomial theorem states that for any positive integer \(k\):
\[ (a + b)^k = \sum_{r=0}^{k} \binom{k}{r} a^{k-r} b^r \]
For our case, we set \(a = 1\) and \(b = x\), and \(k = m+n\):
\[ (1+x)^{m+n} = \sum_{r=0}^{m+n} \binom{m+n}{r} x^r \]

2. **Identify the Coefficients**:
The coefficient of \(x^m\) in the expansion is given by the term where \(r = m\):
\[ \text{Coefficient of } x^m = \binom{m+n}{m} \]
Similarly, the coefficient of \(x^n\) is given by the term where \(r = n\):
\[ \text{Coefficient of } x^n = \binom{m+n}{n} \]

3. **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 \(m\) items from \(m+n\) is equivalent to leaving out \(n\) items, which is the same as choosing \(n\) items from \(m+n\).

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