Step-by-step Solution:

The mean will be: \[ \frac{\sum fx}{\sum f} = \frac{\sum\limits_{n=0}^{50} n \cdot {^{50}C_n}}{\sum\limits_{n=0}^{50} {^{50}C_n}} = \frac{\sum\limits_{n=0}^{50} n \cdot {^{50}C_n}}{2^{50}} \quad \text{(1)} \] We know that: \[ (1 + x)^{50} = {^{50}C_0} + {^{50}C_1}x + {^{50}C_2}x^2 + \cdots + {^{50}C_n}x^n \] Differentiating with respect to \(x\), we have: \[ 50(1 + x)^{49} = {^{50}C_1} + 2 \times {^{50}C_2}x + \cdots + 50 \times {^{50}C_{50}}x^{49} \] Now, put \(x = 1\) in the above equation: \[ 50 \times 2^{49} = {^{50}C_1} + 2 \times {^{50}C_2} + \cdots + 50 \times {^{50}C_{50}} \] Put this value in equation (1): \[ \text{Mean} = \frac{50 \times 2^{49}}{2^{50}} = 25 \] The question can be solved by taking smaller values, e.g., taking the total number as 5. You will find the mean as 2.5. From the choices, we can conclude that the answer is \( \frac{50}{2} = 25 \).