At a large US university student account passwords must be 8

At a large US university, student account passwords must be 8 characters in length. They can contain lower case letters {a, b, . . . , z}, upper case letters {A, B, . . . , Z}, and numbers {0, 1, . . . , 9} only. Each password must contain at least one lower case letter, at least one upper case letter, and at least one number. How many passwords are possible in this system?

Solution

Given at least one lower case one upper case and one has to be selected.

This can be done in 26*26*10 ways

Out of 8 characters 3 are already filled as above

Remaining 5 characters can be in any way.

Total filling options=(26+26+10)=62

5 characters of password be filled with 62 in (62^5) ways

Total no.of ways =6760(62^5)