Solve the PigeonHole Password Problem

[snaidu228]

PigeonHole Help!

Homework Statement

A computer password is formed from 4, 5 or 6 characters. A character is either a lowercase or uppercase vowel: (a, e, i, o, u, y) or (A, E, I, O, U, Y) (passwords are case sensitive) or else it is a digit from the set {0, 3, 4, 7, 9}. Each password must contain at least one digit. How many passwords are possible?

Homework Equations

Product or sum rules

The Attempt at a Solution

12 letters, 5 digits

if 4: 3 letters and 1 digit.
So 3*12= 36 letters possibilities
1*5= 5 digit possibilities

36+5= 41 passwords




I'm not sure if this is how?

 

[Mark44]

Homework Statement

A computer password is formed from 4, 5 or 6 characters. A character is either a lowercase or uppercase vowel: (a, e, i, o, u, y) or (A, E, I, O, U, Y) (passwords are case sensitive) or else it is a digit from the set {0, 3, 4, 7, 9}. Each password must contain at least one digit. How many passwords are possible?

Homework Equations

Product or sum rules

The Attempt at a Solution

12 letters, 5 digits

if 4: 3 letters and 1 digit.
So 3*12= 36 letters possibilities
1*5= 5 digit possibilities

36+5= 41 passwords




I'm not sure if this is how?

This is definitely not how.

Count the numbers of 4-character, 5-character, and 6-character passwords separately. For 4-char passwords, look at separate cases for 1 digit, 2 digits, 3 digits, and 4 digits.
4-character pwds
1 digit + 3 characters: 5 * 12 * 12 * 12 = 8640 possible choices.
2 digits + 2 chars ...
3 digits + 1 char...
4 digits + 0 chars ...

5-character pwds
1 digit + 4 characters:
2 digits + 3 chars ...
3 digits + 2 chars ...
4 digits + 1 char ...
5 digits + 0 char ...

Do the same for 6-character passwords. Add up all the possibilities.

 

[awkward]

Suggestion:
(1) Count all the passwords, ignoring the constraint that they must contain at least one digit.
(2) Then count all the passwords that do not contain any digits.
(3) Subtract (2) from (1) to find the number of passwords that contain at least one digit.