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Im trying to find all combinations of P_{6}. Book solution in paint doc.
My solution: Please tell me where I am going wrong.
P_{6}: 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 P_{61} be defined as follows. C_{i} is the i^{th} character of P_{61}, i = 1,...,6
Let C_{1} be a digit then the following characters can be a digit or uppercase letter.
C_{1} has 10 choices and C_{i} has 36 choices for i = 2,..,6.
Therefore the password defined by P_{61}, which was defined by restricting C_{1} to be a digit, has a total of 10*36^{5} 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.
P_{61} P_{62} ... P_{66}
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 P_{6} and there must be 10*36^{5} choices per sub catagoy.
Therefore total choices for P_{6}
Ʃ(Number of choices for P_{6i}) i = 1,...,6
= Ʃ(10*36^{5}) i = 1,...,6
= 6*10*36^{5}
What am I counting extra of?
My solution: Please tell me where I am going wrong.
P_{6}: 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 P_{61} be defined as follows. C_{i} is the i^{th} character of P_{61}, i = 1,...,6
Let C_{1} be a digit then the following characters can be a digit or uppercase letter.
C_{1} has 10 choices and C_{i} has 36 choices for i = 2,..,6.
Therefore the password defined by P_{61}, which was defined by restricting C_{1} to be a digit, has a total of 10*36^{5} 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.
P_{61} P_{62} ... P_{66}
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 P_{6} and there must be 10*36^{5} choices per sub catagoy.
Therefore total choices for P_{6}
Ʃ(Number of choices for P_{6i}) i = 1,...,6
= Ʃ(10*36^{5}) i = 1,...,6
= 6*10*36^{5}
What am I counting extra of?
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