Question

The domain of the function $f(x)={}^{24-x}{C}_{3x-1}+{}^{40-6x}{C}_{8x-10}$ is ,

• A. {2, 3}
• B. {1 , 2, 3}
• C. {1 , 2, 3 , 4 }
• D. None of these

Answer 1

Answer: (A)

${}^{24-x}{C}_{3x-1}$ is defined if,
$24-x>0,3x-1\ge 0$ and $24-x\ge 3x-1$
$\Rightarrow x<24,x\ge \frac{1}{3}$ and $x\le \frac{25}{4}$
$\Rightarrow \frac{1}{3}\le x\le \frac{25}{4}$
${}^{40-6x}{C}_{8x-10}$ is defined if
$40-6x>0,8x-10\ge 0$ and
$40-6x\ge 8x-10$
$\Rightarrow x<\frac{20}{3},x\ge \frac{5}{4}$ and $x\le \frac{25}{7}$
$\Rightarrow \frac{5}{4}\le x\le \frac{25}{7}$
From (1) and (2), we get $\frac{5}{4}\le x\le \frac{25}{7}$
But 24 - x $\in$ N
$\therefore$ x must be an integer, x = 2, 3.
Hence domain (f) = {2, 3}.