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.
$$\left\{ {1,\,2,\,3,\,4,\,5} \right\}$$
B.
$$\left\{ {2,\,3,\,4,} \right\}$$
C.
$$\left\{ {2,\,3} \right\}$$
D.
none of these
Answer :
$$\left\{ {2,\,3} \right\}$$
Solution :
$$\eqalign{
& 16 - x > 0,\,2x - 1 \geqslant 0,\,16 - x \geqslant 2x - 1,\,20 - 3x > 0,\,4x - 5 \geqslant 0,\,20 - 3x \geqslant 4x - 5 \cr
& \Rightarrow x < 16,\,x \geqslant \frac{1}{2},\,x \leqslant \frac{{17}}{3},\,x < \frac{{20}}{3},\,x \geqslant \frac{5}{4},\,x \leqslant \frac{{25}}{7} \cr
& \Rightarrow \frac{5}{4} \leqslant x \leqslant \frac{{25}}{7} \cr} $$
But $$16 - x\, \in \,N.$$ So, $$x$$ must be integer. Hence, $$x=2,\,3$$