# If a denotes the number of permutations of (x + 2) things taken all at a time,

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 = 182 bc, find the value of x.

Solution:

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

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

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

$a=182 b c$

$\Rightarrow(x+2) !=182 \frac{x !}{(x-11) !} \times(x-11) !$

$\Rightarrow(x+2) !=182(x !)$

$\Rightarrow \frac{(x+2) !}{x !}=182$

$\Rightarrow(x+2)(x+1)=182$

$\Rightarrow(x+2)(x+1)=14 \times 13$

$\Rightarrow x+2=14$

$\Rightarrow x=12$