If a variable takes values 0, 1, 2,…, 50 with frequencies $1,\, {{50}}_{{{C}}_1},{{50}}_{{{C}}_2},\ldots..,{{50}}_{{{C}}_{50}}$, then the AM is

Given: Values \(0,1,2,\dots,50\) with frequencies \(1,\binom{50}{1},\binom{50}{2},\dots,\binom{50}{50}\) (i.e., \(\binom{50}{k}\) for value \(k\)).

Total frequency: $$\sum_{k=0}^{50}\binom{50}{k}=2^{50}.$$

Sum of (value × frequency): $$\sum_{k=0}^{50} k\binom{50}{k}=50\cdot 2^{49}\quad\text{(identity: }\sum k\binom{n}{k}=n2^{\,n-1}\text{)}.$$

Arithmetic Mean (AM): $$\bar{x}=\frac{\sum k\binom{50}{k}}{\sum \binom{50}{k}} =\frac{50\cdot 2^{49}}{2^{50}}=\frac{50}{2}=25.$$

Answer: \(25\).