Square-root part: we need \(\log_{1/4}\left(\frac{5x - x^2}{4}\right) \ge 0\). The base is less than 1, so this means \(0 < \frac{5x - x^2}{4} \le 1\).

• \(5x - x^2 > 0 \Rightarrow 0 < x < 5\)
• \(5x - x^2 \le 4 \Rightarrow x^2 - 5x + 4 \ge 0 \Rightarrow (x-1)(x-4) \ge 0 \Rightarrow x \le 1\) or \(x \ge 4\)

Together: \(x \in (0, 1] \cup [4, 5)\).

\({}^{10}C_x\) part: \(x\) must be an integer from 0 to 10.

The integers in \((0,1] \cup [4,5)\) are 1 and 4.

Domain = {1, 4}

Answer: (C)