In the expansion of $(1+x)^{50}$, the sum of the coefficients of odd powers of x is:

Question

Accepted Answer

To determine the sum of the coefficients of the odd powers of $ x $ in the expansion of $ (1+x)^{50} $, we can use a well-known algebraic technique involving the evaluation of the polynomial at specific values.

### Explanation for Option 3:

1. **Sum of Coefficients of All Terms:**
   The sum of the coefficients of all terms in the expansion of $ (1+x)^{50} $ is obtained by substituting $ x = 1 $ into the expression. This gives:
   $$
   (1+1)^{50} = 2^{50}
   $$

2. **Sum of Coefficients of Even Powers:**
   Similarly, the sum of the coefficients of the even powers of $ x $ is obtained by substituting $ x = -1 $ into the expression. This gives:
   $$
   (1+(-1))^{50} = 0^{50} = 0
   $$

3. **Sum of Coefficients of Odd Powers:**
   The sum of the coefficients of the odd powers of $ x $ can be found by subtracting the sum of the coefficients of the even powers from the sum of all coefficients:
   $$
   \text{Sum of odd coefficients} = 2^{50} - 0 = 2^{50}
   $$

However, this is not the final answer. The sum of the coefficients of the odd powers is actually half of this value because the expansion alternates between even and odd powers. Therefore, the sum of the coefficients of the odd powers is:
   $$
   \frac{2^{50}}{2} = 2^{49}
   $$

Thus, the correct answer is $ 2^{49} $, which corresponds to Option 3.

### Explanation for Why Other Options Are Incorrect:

- **Option 1 (0):** This option suggests that the sum of the coefficients of the odd powers is zero. This would only be true if the polynomial had no odd power terms, which is not the case here.

- **Option 2 (1):** This option implies that the sum of the coefficients of the odd powers is 1, which is incorrect because the calculation clearly shows that the sum is $ 2^{49} $.

- **Option 4 ($2^{51}$):** This option suggests a sum larger than the total sum of all coefficients, $ 2^{50} $, which is not possible. The sum of the coefficients of the odd powers cannot exceed the sum of all coefficients.

Therefore, Option 3 is the correct answer, as it accurately reflects the sum of the coefficients of the odd powers in the expansion of $ (1+x)^{50} $.