How do you know that 7 appears 11000 times between 1 1000?

Answer 1

You have to break the problem up into parts. This answer is going to assume only real numbers (for obvious reasons).

First look at the ones place: 0007 to 9997. The ones place gets cycled through a total of 1000/10 times, for a total of 100 sevens.

Now look at the tens place: 0070 to 9979. The tens place gets cycled through a total of 1000/100 times, but each cycle has 10 sevens. 10*10 = 100 more sevens.

Now look at the hundreds place: 0700 to 0799. The hundreds place only occurs once, but there are a hundred numbers between 700 and 799, giving another 100 sevens.

First look at the numbers 1-99. It appears at 7, 17, 27, 37, 47, 57, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 87, and 97 for a total of 20 times (because 77 counts twice).

So the total is 300, not 11000. There aren't even that many numbers between 1 and 1000.