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. :)