Question

The domain of the function $f\left(x\right)={}^{16-x}{C}_{2x-1}+{}^{20-3x}{P}_{4x-5},$ where the symbols have their usual meanings, is the set:

• A. $\{1,2,3,4,5\}$
• B. $\{2,3,4\}$
• C. $\{2,3\}$
• D. None of the above

Answer 1

Answer: (C)

$\{2,3\}$

${}^n{C}_r$ is defined when $n,r\in N,n\ge r$ ($r$ can be zero)
${}^{16-x}{C}_{2x-1}$ is defined for
$16-x>\mathrm{0.....}\left(1\right)2x-1\ge \mathrm{0.....}\left(2\right)16-x\ge 2x-\mathrm{1.....}\left(3\right)$
Combining $\left(1\right),\left(2\right)$ and $\left(3\right)$ we get
.
$x=\{1,2,3,4,5\}$.........$\left(i\right)$
.
Similarly, ${}^{20-3x}{P}_{4x-5}$ is defined for
$20-3x>\mathrm{0....}\left(4\right)4x-5\ge \mathrm{0....}\left(5\right)20-3x\ge 4x-\mathrm{5.....}\left(6\right)$
.
Combining $\left(4\right),\left(5\right)$ and $\left(6\right)$ we get
.
$x=\{2,3\}$.........$\left(ii\right)$
.
From $\left(i\right)$ and $\left(ii\right)$, the common values of $x$ are $2$ and $3$.