Step 1: Understanding the Concept:
The function involves permutations, denoted as $^n P_r$. For this mathematical operation to be valid, specific conditions regarding $n$ and $r$ must be met.
Step 2: Key Formula or Approach:
For $^n P_r$ to be defined:
1. $n$ must be a non-negative integer ($n \ge 0, n \in \mathbb{Z}$).
2. $r$ must be a non-negative integer ($r \ge 0, r \in \mathbb{Z}$).
3. $n$ must be greater than or equal to $r$ ($n \ge r$).
Step 3: Detailed Explanation:
Given the function $f(x) = {^{7-x}}P_{x-1}$, we identify $n = 7-x$ and $r = x-1$.
Let's apply the conditions:
Condition 1: $n \ge 0$
\[ 7 - x \ge 0 \implies x \le 7 \]
Condition 2: $r \ge 0$
\[ x - 1 \ge 0 \implies x \ge 1 \]
Condition 3: $n \ge r$
\[ 7 - x \ge x - 1 \]
\[ 8 \ge 2x \implies x \le 4 \]
Now, we find the intersection of these inequalities:
From (1) and (2) and (3), we must have $1 \le x \le 4$.
Additionally, for permutations, $n$ and $r$ must be integers.
$7-x$ and $x-1$ are integers if and only if $x$ is an integer.
The integer values of $x$ in the interval $[1, 4]$ are $1, 2, 3, 4$.
Therefore, the domain is the discrete set $\{1, 2, 3, 4\}$.
Step 4: Final Answer:
The domain is $\{1, 2, 3, 4\}$.