Quandaries and Queries


I had a discussion with my father about how many four digit numbers you can make with 0-9. He was saying that there is 12,000 and that seems quite high for me. But can you tell me how many there are, and what exactly are they?

Thank you



Hi David,

It depends what you mean by a four digit number. Is 0001 a four digit number or, since 0001 = 1, is it a one digit number? What about 0000 or 0104?

If the first digit can't be zero (so that 0001, 0000 and 0104 are not four digit numbers) then the smallest four digit number is 1,000 and the largest is 9,999. In this situation there are 9,000 four digit numbers.

If a four digit number is any string of four characters from 0-9 (so that 0001 and 0000 are four digit numbers) then we can reason as follows.

Start with the first digit. There are 10 possible choices fro the first digit, 0, 1, 2,..., 9

Regardless of what number you chose for the first digit, there are 10 possible choices for the second digit. Thus there are 10 10 = 102 = 100 possible two digit numbers.

Now for each of the 100 possible choices of a two digit number you can append any of the ten digits to make a three digit number. Thus there are 100 10 = 103 = 1000 possible three digit numbers.

In a similar way there are 104 = 10,00 possible four digit numbers.