The domain of $f(x)=\sqrt{\log_{1/4}\left(\frac{5x-x^2}{4}\right)}+{^{10}C}_x$ is

{1, 4}

Let $f_1=\sqrt{\log_{1/4}\left(\frac{5x-x^2}{4}\right)}$ and $f_2={^{10}C}_x$

Clearly $f_1$ is defined for $\log_{1/4}\left(\frac{5x-x^2}{4}\right)≥0$

$⇒0<\frac{5x-x^2}{4}≤1⇒\frac{5x-x^2}{4}>0$ and $\frac{5x-x^2}{4}≥1$

⇒ x(x - 5) < 0 and $x^2-5x+4≥0$ ⇒ x ∈ (0, 5) and  x ∈ (-∞, 1] ∪ [4, ∞)

⇒ $f_1$ is defined for x ∈ (0, 1] ∪ [4, 5)

$f_2$ is defined for x ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

f(x) is defined for $x ∈ D_{11} ∩ D_{12} = \{1, 4\}$