Solve the P6 Counting Problem: Unlock 10*365 Password Combinations

[Miike012]

Im trying to find all combinations of P₆. Book solution in paint doc.

My solution: Please tell me where I am going wrong.

P₆: Password of 6 characters
1. Each password must contain at least one digit,
2. Each character of password can be a digit or uppercase letter.

Let P₆₁ be defined as follows. C_i is the i^th character of P₆₁, i = 1,...,6

Let C₁ be a digit then the following characters can be a digit or uppercase letter.
C₁ has 10 choices and C_i has 36 choices for i = 2,..,6.

Therefore the password defined by P₆₁, which was defined by restricting C₁ to be a digit, has a total of 10*36⁵ choices.

I will do the same for P_6i, i = 2,...6, where the i^th character is a digit.

All passwords P_6i i = 1,2,3,4,5,6 will look like the following:
Let D represent the character that is a digit and DL represent the character that is a digit or uppercase letter.

P₆₁ P₆₂ ... P₆₆
1.D 1.DL 1.DL
2.DL 2.D 2.DL
3.DL 3.DL 3.DL
4.DL 4.DL 4.DL
5.DL 5.DL 5.DL
6.DL 6.DL 6.D


Hence you can see that there are 6 total different "sub catagories" of P₆ and there must be 10*36⁵ choices per sub catagoy.
Therefore total choices for P₆
Ʃ(Number of choices for P_6i) i = 1,...,6
= Ʃ(10*36⁵) i = 1,...,6
= 6*10*36⁵



What am I counting extra of?

 

[Miike012]

deleted

 

[verty]

A password like 11XXXX will appear in group 1 and group 2.

 

[haruspex]

How many passwords satisfy (2). How many of those do not satisfy (1)?