Among the positive integers less than , do the majority contain the digit 1?
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As is often the case, it's easier to count the integers below 1 0 0 0 0 0 0 0 which do not contain the digit 1 .
This leaves 9 possibilities for each digit: 0 and 2 to 9 .
So, for the integers up to 9 9 9 9 9 9 9 , we have 9 7 − 1 = 4 7 8 2 9 6 8 integers which do not contain the digit 1 . We had to subtract 1 because we had counted 0 as one of our possibilities.
Therefore, 9 9 9 9 9 9 9 − 4 7 8 2 9 6 8 = 5 2 1 7 0 3 1 integers contain the digit 1 , which is the majority.