Question

All the $7-$digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once, and not divisible by $5$, are arranged in increasing order. Then ${1897}^{\text{th}}$ number in the list is

• A. $4276531$
• B. $4213567$
• C. $4176532$
• D. $4123567$

Answer 1

The correct option is B $4213567$

The $7-$digit number with $1$ in the left most place and containing each of the digits $1,2,3,4,5,6,7$ exactly once is $6!$

But $5!$ numbers out of those $6!$ have $5$ at the end that are divisible by $5.$
Hence, number of numbers starting with $1\_\_\_\_\_\_=6!-5!=720-120=600$
Similarly, required number of numbers starting with $2\_\_\_\_\_\_=600$
and number of numbers starting with $3\_\_\_\_\_\_=600$
Total $=1800$

Number of numbers starting with $41\_\_\_\_\_=120-24=96$
Total $=1800+96=1896$

So, ${1897}^{\text{th}}$ number is $4213567$.