Question

If $a$ denotes the number of permutations of $x+2$ things taken all at a time, $b$ the number of permutations of $x$ things taken $11$ at a time and $c$ the number of permutations of $x-11$ things taken all at a time such that $a=182bc$, then the value of $x$ is

• A. $15$
• B. $12$
• C. $10$
• D. $18$

Answer 1

The correct option is B $12$

$a$ denotes the number of permutations of $(x+2)$ things taken all at a time $=(x+2)!$
$b$ the number of permutations of $x$ things taken 11 at a time $={}^x{P}_{11}$
And $c$ the number of permutations of $(x-11)$ things taken all at a time $=(x-11)!$
Given : $a=182bc$
$\Rightarrow (x+2)!=182\times {}^x{P}_{11}\times (x-11)!$
$\Rightarrow (x+2)!=182\times \frac{x!}{(x-11)!}\times (x-11)!$
$\Rightarrow (x+2)(x+1)=182$
$\Rightarrow x^2+3x-180=0$
$\Rightarrow (x+15)(x-12)=0$
It cannot be $-15$
So, $x=12$
Hence, option 'B' is correct.