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

The Correct Option is A

$^{24-x}C_{3x-1}$ is defined if, $ 24-x > 0, 3x-1 \ge0 $ and $24-x \ge3x-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 \ge8x -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}.