Video Explanation

Text Explanation

For this (and all) counting problems, we need to ask if order matters. In this password question, we essentially need to know if these two 9-digit long passwords are different: 

123456789 vs. 987654321. 

Each has the same 9 digits, but arranged differently. Well, as we know from PIN numbers and email passwords, these two values are different passwords. Hence, order matters and we’re dealing with a permutation question. 

Great! Our next step is to tackle the at least 9-digits long part of our question. First, let’s list all the possible digits. They are: 

0, 1, 2, 3, 4, 5, 6, 7, 8, 9. 

In total, there are 10 digits from 0 - 9. Since no digit may repeat, and the password must be at least 9-digits long we have only two possible lengths: 9 digits long, or the full 10 digits long password. 

For the 9-Digits long password, we lay out 9 slots: 

_  _  _  _  _  _  _  _  _ .  

In the first slot, we have 10 digits to choose from; in the second, we have 9 digits to choose from, and so on. So, we’re left with: 

9-Digits = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2

This is the same as 10P9 = 10! 

For the 10-Digits long password, we lay out 10 slots:

_  _  _  _  _  _  _  _  _  _ .  

Again, in the first slot, we have 10 digits to choose from; in the second, we have 9 digits to choose from, and so on until we fill the 10th slot. We have: 

10-Digits = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1. 

This is the same as 10P10 = 10! 

Finally, we add these two distinct sets of possible password arrangements together to have 10! + 10! = 2 x 10!  passwords. Voila! That is our answer :) 

Frequently Asked Questions

FAQ: Just to clarify—can the first number in this password be 0?

A: Yes, the first digit CAN be 0. If we changed from numbers to letters, any single letter could be the start of our password. In the same way, 0 is allowed to be the first digit in a string of digits for a password even though if we were simply making a number we wouldn't include it. This isn't quite a number —it's a password. On your computer, it's possible to type in a number starting with 0 as your password, so we can do it here as well. :)