Question

The domain of the function $f\left(x\right)={}^{21-x}C_{3x-1}+{}^{25-3x}P_{5x-3}$ is

• A. $\left\{1,2,3\right\}$
• B. $\left\{2,3\right\}$
• C. $\left\{2,3,4\right\}$
• D. $\left\{2,3,4,5\right\}$

Answer 1

The correct option is A

$\left\{1,2,3\right\}$

Explanation for the correct options:

Find the domain of function $f\left(x\right)$:

The given function is:

$f\left(x\right)={}^{21-x}C_{3x-1}+{}^{25-3x}P_{5x-3}$

The above function is defined as:

${}^{n}C_r$is defined when $n,r\in N,n\ge r(r\text{can be zero})$

${}^{21-x}C_{3x-1}$​ is defined for
$$\begin{gathered} \Rightarrow 21-x>0 \\ \Rightarrow x<21\dots \left(1\right) \\ \mathrm{And} \\ 3x-1\ge 0 \\ \Rightarrow x\ge \frac{1}{3}\dots \left(2\right) \\ \mathrm{And} \\ \Rightarrow 21-x\ge 3x-1 \\ \Rightarrow 4x\le 22 \\ \Rightarrow x\le \frac{11}{2}\dots \left(3\right) \end{gathered}$$

Similarly, ${}^{25-3x}P_{5x-3}$ is defined for

$$\begin{gathered} \Rightarrow 25-3x\ge 0 \\ \Rightarrow x\le \frac{25}{3}\dots \left(4\right) \\ And \\ \Rightarrow 5x-3\ge 0 \\ \Rightarrow x\ge \frac{3}{5}\dots \left(5\right) \\ And \\ \Rightarrow 25-3x\ge 5x-3 \\ \Rightarrow 8x\le 28 \\ \Rightarrow x\le \frac{7}{2}\dots \left(6\right) \end{gathered}$$

Combing all $6$ equations(taking the intersection of all the six equations) we get,

$\frac{3}{5}<x\le \frac{7}{2}$

Since, the domain of the function $f\left(x\right)$ is $\left\{1,2,3\right\}$

Hence, the correct answer is option A.