Xavier Aptitude Test 2011 Paper

(c) According to the question a number is a magic numberv if it satisfies any one of the following conditions:
(a) ${\mathit{x}}_1{\mathit{x}}_2{\mathit{x}}_3={\mathit{x}}_4{\mathit{x}}_5{\mathit{x}}_6\ne {\mathit{x}}_5{\mathit{x}}_6{\mathit{x}}_7$
(b) ${\mathit{x}}_1{\mathit{x}}_2{\mathit{x}}_3={\mathit{x}}_5{\mathit{x}}_6{\mathit{x}}_4\ne {\mathit{x}}_4{\mathit{x}}_5{\mathit{x}}_6$
(c) ${\mathit{x}}_1{\mathit{x}}_2{\mathit{x}}_3={\mathit{x}}_4{\mathit{x}}_5{\mathit{x}}_6={\mathit{x}}_5{\mathit{x}}_6{\mathit{x}}_7$
Consider $($ C $){\mathit{x}}_4{\mathit{x}}_5{\mathit{x}}_6={\mathit{x}}_5{\mathit{x}}_6{\mathit{x}}_7$
implies ${\mathit{x}}_4={\mathit{x}}_5,{\mathit{x}}_5={\mathit{x}}_6,{\mathit{x}}_6={\mathit{x}}_7$
$\Rightarrow {\mathit{x}}_1={\mathit{x}}_2={\mathit{x}}_3={\mathit{x}}_4=............={\mathit{x}}_7$
There are 9 such numbers.
The different types of roots and the corresponding number of magic numbers are given below.For $\left(1\right)$ a can have any value from $1$ to $9.$
$\left(2,3\right)$ a can have any value from $0$ to $9$ and b can have $9$ different values.
$\left(4,5\right)$ ${\mathit{x}}_1\mathrm{}{\mathit{x}}_2\mathrm{}{\mathit{x}}_3$ can have $990\left(999-9\right)$ values and the blank can be filled in $10$ ways.