Question

If $n$ is the natural number.

Then, the range of the function $f\left(n\right)={}^{8-n}{P}_{n-4},4\le n\le 6$, is

• A. $\{1,2,3,4\}$
• B. $\{1,2,3,4,5,6\}$
• C. $\{1,2,3\}$
• D. $\{1,2,3,4,5\}$
• E. $\varnothing$

Answer 1

The correct option is C

$\{1,2,3\}$

Explanation for correct answer

$f\left(n\right)={}^{8-n}{P}_{n-4}$

Therefore the domain is

$8–n\ge 0$and $n–4\ge 0$

$\Rightarrow$$8\ge n$ and $n\ge 4$

$\Rightarrow$$n–4\le 8–n$

$\Rightarrow$ $2n\le 12$

$\Rightarrow$ $n\le 6$

If $n=4,{}^4{P}_0=1$

If $n=5,{}^3{P}_1=3$

If $n=6,{}^2{P}_2=2$

The range of $f\left(n\right)$ is $\{1,2,3\}$.

Hence, option C is correct.