Question

Let a seven digit number be formed using $1,2,3,4,5,6,7$ without repetition such that they are not divisible by $5$. If the numbers are arranged in increasing order, then the ${1994}^{th}$ number is

Answer 1

The number of numbers whose first digit is $1$
$=6!-5!=720-120=600$
Similarly, first digit is $2$
$=600$
Similarly, first digit is $3$
$=600$
The total numbers till now $=1800$
Now, first digit is $4$ and second digit is $1$, then
$5!-4!=120-24=96$
The total numbers till now $=1896$
First digit is $4$ and second digit is $2$, then
$=96$
The total numbers till now $=1896+96=1992$
Therefore, ${1993}^{th}\text{term}\to 4312567$
${1994}^{th}\text{term}\to 4312576$