Indian Regional Mathematical Olympiad Problem 1
Probability
Level
2
Consider all the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once and that are not divisible by 5. They are arranged in a list in increasing order. Find the 2000th number in this list.
The answer is 4315672.
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1 solution
The number of 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 these end in 5 and hence are divisible by 5. Thus the number of 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 but not divisible by 5 is 600. Similarly the number of 7-digit numbers with 2 and 3 in the left most place and containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once but not divisible by 5 is also 600 each. These account for 1800 numbers. Hence 2000-th number must have 4 in the left most place. Again the number of such 7-digit numbers beginning with 41,42 and not divisible by 5 is 120 − 24 = 96 each and these account for 192 numbers. This shows that 2000-th number in the list must begin with 43. The next 8 numbers in the list are: 4312567, 4312576, 4312657, 4312756, 4315267, 4315276, 4315627 and 4315672. Thus 2000-th number in the list is 4315672.