Math CentralQuandaries & Queries


Question from swathi, a student:

How many 4 digit numbers can be formed using the digit 3 only once but the digit 3 must be used in all numbers?

Hi Swathi,

Suppose the 3 is in the leftmost place, that is 3 is the thousandth digit. You now have 9 choices for the hundredth digit, 0, 1, 2, 4, 5, 6, 7, 8 or 9. You also have 9 choices for the tens digit and for the units digit and thus if 3 is the thousandths digit you can form $9 \times 9 \times 9 = 9^3$ numbers.

Now suppose that 3 is the hundredths digit. How many choices do you have for the thousandths digit? Well is depends. Is 03456 a 4 digit number? If so you have 9 choices for the thousandths digit, if not you only have 8 choices.

Can you complete the problem from here?

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