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Q.
The domain of the function $f(x) = {^{24- x}C_{3x-1}} + {^{40 -6x}C_{8x-10}} $ is ,
Relations and Functions - Part 2
Solution:
$^{24-x}C_{3x-1}$ 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}.