Question

All the five digits (formed rom $1,\mathrm{2...9}$) number in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The $97th$ number in the list does not contain the digit

• A. $4$
• B. $5$
• C. $7$
• D. $8$

Answer 1

The correct option is B $5$

The rank will start if number start form $1.$

A number of $1x$ can be formed by taking last $4$ digits from $8$

$\Rightarrow$ So, number of ways $={}^8{C}_4=70$

and for $2x$ it ${}^7{C}_4=35.$

$\Rightarrow$ Of the form $23\underset{–}{x}$ is ${}^6{C}_3=20$

$\Rightarrow$ and of form $24\underset{–}{x}$ is ${}^5{C}_3=10$

$\therefore$ For $97th$ number $1st$ two digit are $24$

$\Rightarrow$ Now of form $245x$ is ${}^4{C}_2=6$

$\Rightarrow$ i.e. its rank $70+20+6=96$

or $97th$ number is of form $246x$ is $\underset{–}{7}\underset{–}{8}$

$\therefore$ Number is $24678.$