The domain of the function $f(x) = {}^{(10-x)}C_{(x-6)}$ is...

For the function $f(x) = {}^{(10-x)}C_{(x-6)}$ to be defined, the following conditions must be satisfied:

$10 - x \ge 0 \Rightarrow x \le 10$

$x - 6 \ge 0 \Rightarrow x \ge 6$

$10 - x \ge x - 6 \Rightarrow 2x \le 16 \Rightarrow x \le 8$

Also, $10 - x$ and $x - 6$ must be integers, which implies $x$ must be an integer.

Combining the inequalities, we get $6 \le x \le 8$.

Since $x$ is an integer, the possible values of $x$ are $6, 7, 8$.

Thus, the domain of the function is $\{6, 7, 8\}$.

Answer: $\{6, 7, 8\}$