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 which of the following is/are CORRECT ?

• A. $1993^{\text {rd }}$ number in the list is 4312567
• B. $1996^{\text {th }}$ number in the list is 4312756
• C. $2000^{\text {th }}$ number in the list is 4315672
• D. $1999^{\text {th }}$ number in the list is 43152763

Answer 1

Answer: (A), (B), (C)

The 7-digit numbers with 1 in the left most place and containing each of the digits $1, 2,3, 4, 5, 6, 7$ exactly once is $6! = 720.$
But 120 of there end in 5 and hence are divisible by 5

hence, 2000-th number must have 4 in the left most place.
Again the numbers of such 7-digit numbers beginning with 41,42 and not divisible by 5 is $120-24=96$ each and there account for 192 numbers.
2000-th number in the list beginning with 43
next 8 numbers in the test are :
$4312567,4312576,4312657,4312756,4315267,5315276,4315627,4315672 .$