Bunuel wrote:
All the five digit numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The 97th number in the list does not contains the digit
(A) 4
(B) 5
(C) 7
(D) 8
(E) 9
We start with the number 12345. We can increase the unit digit from 5 to 9, so that is 5 numbers. The next 5-digit number in order would be 12356, we can increase the unit digit from 6 to 9 so that is 4 numbers. We can keep doing this until we need to change the hundreds digit.
We can see the pattern eventually turns into 5 + 4 + 3 + 2 + 1 = 15 and the 16th number we hit is 12456. After we increase the hundreds digit, look at the last two digits 56 which we had earlier from 12356. So if we want to hit the next change in hundreds, 12567, we need to add 4+3+2+1 = 10. Then we can again repeat this pattern, 15 + 10 + 6 + 3 + 1 = 35 to hit 13456.
13456 is the first number where we change the thousands digit, we used the numbers 15 + 10 + 6 + 3 + 1 to achieve that. Now we can repeat what we did earlier for each change in thousands, 14567 would have 35 + (10 + 6 + 3 + 1) = 55 numbers before it. 15678 would have 55 + (6 + 3 + 1) = 65 numbers before it.
Again repeat the numbers above, to hit 23456, we need to add 65 + (3 + 1) + 1 = 70. We did 13456 to 14567 earlier which needed 20 numbers, so to hit 24567 we need 20 more numbers too. Then 24567 has 70+20 = 90 numbers before it (#91 in list). We can count our way up now to find #96 is 24689 then #97 is 24789.
Ans: B
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