A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

Question: A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?

Solution:
There are 10 digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
No digit may be repeated.
Each password must be at least 9 digits long. So, possible lengths are,
9 digits or 10 digits

For 9-digit passwords:
Number of ways = ¹⁰P₉ = 10!

For 10-digit passwords:
Number of ways = ¹⁰P₁₀ = 10!

∴ Total number of passwords = 10! + 10!
= 2 × 10!

Correct Answer: খ) 2 × 10!

Note: Since it is a password, 0 can be used as the first digit.