Tony
Father of Math student

Think of it this way: How many unique one digit numbers can be generated using the digits 0-9 (with no other restrictions)? Clearly, there are ten: "0", "1", "2", ... , "9".

Now how many unique two digit numbers can be generated with the same digits? That depends on what you mean by a two digit number. Is 03 a two digit number? If you allow the first digit to be zero then the two digit numbers are 00 to 99. So there are 100 unique numbers. If the first digit can't be zero then the 10 numbers, 00 to 09 are not two digit numbers so the number of two digit numbers is 100 - 10 = 90.

What about three digit numbers? Again if you allow the first digit to be zero then you can turn each of the 100 two digit numbers into 10 three digit numbers by placing one of the digits 0 to 9 in front of the two digit number. For example 12 would generate 012, 112, 212, ... to 912. Thus there are 10 100 = 1000 three digit numbers. If the first digit can't be zero then you need to remove 000 to 099 from your list leaving you with 1000 - 99 = 900 two digit numbers.

What about four and five digit numbers?

Stephen and Penny