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 permutaions of (X-11) things taken all at a time such that a=182bc, find the value of x??

Answer 1

It is given that,

$a={}^{x+2}P_{x+2}=\left(x+2\right)!$

$b={}^{x}P_{11}=\frac{x!}{\left(x-11\right)!}$

$c={}^{x-11}P_{x-11}=\left(x-11\right)!$

Also, a=182bc

$$\begin{gathered} \Rightarrow \left(x+2\right)!=182.\frac{x!}{\left(x-11\right)!}.\left(x-11\right)! \\ \Rightarrow \left(x+2\right)\left(x+1\right)=182 \end{gathered}$$

$$\begin{gathered} \Rightarrow x^2+3x-180=0 \\ \Rightarrow \left(x-12\right)\left(x+15\right)=0 \\ \Rightarrow x=12,or,x=-15 \end{gathered}$$