Question

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

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

Answer 1

Answer: (A)

$\{2,3\}$

${}^{24-x}{C}_{3x-1}$ 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 $x\le \frac{25}{4}\Rightarrow \frac{1}{3}\le x\le \frac{25}{4}$ ...(1)

${}^{40-6x}{C}_{8x-10}$ is defined if

$40-6x>0,8x-10\ge 0$ and $40-6x\ge 8x-10$

$x<\frac{20}{3},x\ge \frac{5}{4}$ and $x\le \frac{25}{7}\Rightarrow \frac{5}{4}\le x\le \frac{25}{7}$ ...(2)

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,

$\therefore x=2,3.$

Hence domain of $f\left(x\right)=2,3$