Question

The decimal parts of the logarithm of two numbers taken at random are found to six place. Probability that second can be subtracted first one without borrowing is

• A. ${\left(\frac{9}{20}\right)}^6$
• B. ${\left(\frac{1}{2}\right)}^6$
• C. ${\left(\frac{11}{20}\right)}^6$
• D. ${\left(\frac{3}{20}\right)}^6$

Answer 1

Answer: (A), (C)

${\left(\frac{9}{20}\right)}^6$
${\left(\frac{11}{20}\right)}^6$

$P_1=x_1{x}_2{x}_3...x_6{P}_2=y_1{y}_2...y_6$
Any digit after decimal can be $0,1,2,...,9$
$P_2$ can be subtracted from $P_1$ without borrowing, if $x_i\ge y_i$
Let $x_i\ge y_i=\lambda (0\le \lambda \le 9)\Rightarrow x_i=\lambda ,\lambda +1,...,9$
$\Rightarrow$ After choosing $y_i,x_i$ can be chosen in $(10-\lambda )$ ways.
$\therefore$ Required probability $={\left(\frac{{\sum }_{\lambda =0}^9(10-\lambda )}{100}\right)}^6={\left(\frac{11}{20}\right)}^6$