Good afternoon from the States.

How many 2-digit numbers do NOT contain any repeating digits? Since a two-digit number is not going to begin in zero, there are 9 permissible values for the first digit, namely one through 9. Zero is allowed for the second digit, but remember that the second digit cannot be the same as the first. Thus, there are 9 permissible values for the second digit. There are therefore 9*9=81 two-digit numbers in which both digits are unique.

What about 3-digit numbers? Same concept: There are 9 permissible values for the first digit (all but zero), 9 permissible values for the second digit (all but the first digit), and 8 permissible values for the third digit (all but the first and second digits). Thus, the answer for the 3-digit case is 9*9*8=648.

Four digits follows similarly: 9*9*8*7.

Clearly, 10 digits is the maximum if you need all digits to be unique. In that case, you get 9*9*8*7*6*5*4*3*2*1. Note that the cases for 9 digits and ten digits are identical.

Hopefully this helps. Please ask if you have any further questions.

-Andy