Assertion (A) : In the `(1+x)^50`, the sum of the coefficients of odd powers of x is `2^49`
Reason ( R) : The sum of coefficients of odd powers of x in `(1+x)^n` is `2^(n-1)`

Options

A. A and R are true , R is correct explanation of A
B. A and R are true, R is not the correct explanation of A
C. A is true , R is false
D. A is false R is true

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question regarding the binomial expansion of \((1+x)^{50}\).

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The binomial expansion of \((1+x)^{50}\) is given by:
\[ (1+x)^{50} = \sum_{k=0}^{50} \binom{50}{k} x^k \]
Here, \(\binom{50}{k}\) represents the coefficients of the expansion.

2. **Identifying Coefficients of Odd Powers**:
The coefficients of odd powers of \(x\) in the expansion are \(\binom{50}{1}, \binom{50}{3}, \binom{50}{5}, \ldots, \binom{50}{49}\).

3. **Finding the Total Sum of Coefficients**:
To find the sum of the coefficients of odd powers, we can evaluate the expression at \(x=1\):
\[ (1+1)^{50} = 2^{50} \]
This represents the sum of all coefficients (both odd and even).

4. **Separating Odd and Even Coefficients**:
Let \(S\) be the sum of the coefficients of odd powers and \(E\) be the sum of the coefficients of even powers. We have:
\[ S + E = 2^{50} \]

5. **Using the Negative Value**:
Now, evaluate the expression at \(x=-1\):
\[ (1-1)^{50} = 0 \]
This gives:
\[ S - E = 0 \]
Thus, we can conclude that:
\[ S = E \]

6. **Solving for S**:
From \(S + E = 2^{50}\) and \(S = E\), we can substitute \(E\) with \(S\):
\[ S + S = 2^{50} \implies 2S = 2^{50} \implies S = 2^{49} \]
Therefore, the sum of the coefficients of odd powers of \(x\) in \((1+x)^{50}\) is indeed \(2^{49}\).

7. **Conclusion**:
The assertion (A) is true: The sum of the coefficients of odd powers of \(x\) in \((1+x)^{50}\) is \(2^{49}\).
The reason (R) is also true: The sum of coefficients of odd powers of \(x\) in \((1+x)^n\) is \(2^{n-1}\).

### Final Answer:
Both assertion (A) and reason (R) are true, and (R) is the correct explanation of (A).