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:
$a={ }^{x+2} P_{x+2}=(x+2) !$
$b={ }^{x} P_{11}=\frac{x !}{(x-11) !}$
$c={ }^{x-11} P_{x-11}=(x-11) !$
$a=182 b c$
$\Rightarrow(x+2) !=182 \frac{x !}{(x-11) !} \times(x-11) !$
$\Rightarrow(x+2) !=182(x !)$
$\Rightarrow \frac{(x+2) !}{x !}=182$
$\Rightarrow(x+2)(x+1)=182$
$\Rightarrow(x+2)(x+1)=14 \times 13$
$\Rightarrow x+2=14$
$\Rightarrow x=12$
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