Math CentralQuandaries & Queries


Question from Alberto, a student:

How many four-digit numbers are there which contain at least one digit 3?

Hello Alberto.

Let me ask you two questions which will make it easier to solve this question:

  1. How many four-digit numbers are there?
  2. How many four-digit numbers are there that have no digit 3 in them?

Clearly, you can subtract one from the other and get the answer to your original question.

You didn't say whether you think of a number with leading zeroes as a possible four digit number. If you assume that a number like 0038 is a four-digit number, then there are 10 possible choices for the first digit, 10 for the second, etc. So there are 10x10x10x10=104 possibilities. That's the answer to the first question. If leading zeroes aren't allowed, then you have to eliminate all the numbers below 1000. So there would be 104 - 1000 choices. You will have to decide for yourself whether "four digit number" means no leading zeroes.

You first have to decide if leading zeroes are permitted (is 0038 a four-digit number?)

To find the answer to the second question, just think about how many possible digits there are for each of the four places (there are nine, because the "3" isn't allowed). There are still four places though, so it's the same idea as for question 1, but with 9 choices for each digit instead of 10. Again, you'll need to correct for leading zeroes here if you don't want to include them.

Hope this helps,
Stephen La Rocque.

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