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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 the increasing order. Find the (2000)th number in this list.

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Solution

The no. of 7 digit no. with 1 in the left most place and containing each of the digit 1,2,37 exactly is 6!=720,
But 120 of these end in 5 and hence are divisible by 5. There the no. of 7 digit no. with 1 in the left most place and containing each of the digit containing 1,2,37 exactly once but no divisible by 5 is 600.
Similarly the no. of 7 digit no. with 2 & 3 in the left most place and containing each of the digit 1,2,37 exactly once but not divisible by 5 is also 600 each.
Hence 2000th no. must have 4 in the left most place. Again the no. of such 7 digit no. begin with 41,42 and not divisible by 5 is 12024=96 each and the account for 192 no.'s.
These show that, 2000th no. must begin with 43 and the next 8 no. in the list are, 4312567,4312576,4312657,4312756,4315267,4315276,4315627,4315672.

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