Question

The domain of the function $f(x) = {^{7-x}\text{P}_{x-1}$ is

• A. $\mathbb{R}$
• B. $x \in \mathbb{R} - \{1\}$
• C. $\{1, 2, 3, 4\}$
• D. $\{1, 2, 3, 4, 5, 6\}$

Hint:

In permutations/combinations, the upper index must be greater than or equal to the lower index, and both must be non-negative integers.

Answer 1

The Correct Option is C

Step 1: Concept
For ${^n}\text{P}_r$ to be defined, $n$ and $r$ must be non-negative integers such that $n \ge r$.

Step 2: Meaning
We have three conditions: $7-x \in \mathbb{Z}^+$, $x-1 \in \mathbb{Z}_{\ge 0}$, and $7-x \ge x-1$.

Step 3: Analysis
1) $7-x > 0 \implies x < 7$. 2) $x-1 \ge 0 \implies x \ge 1$. 3) $7-x \ge x-1 \implies 8 \ge 2x \implies x \le 4$. Since $x$ must be an integer, $x \in \{1, 2, 3, 4\}$.

Step 4: Conclusion
The domain is the set of integers $\{1, 2, 3, 4\}$. Final Answer: (C)