Order with Repetition Counting problem

Each part is a little different. Where are you having trouble?

I don't think of this sort of problem in terms of a list of known steps to take. You may be able to identify a problem as one of combinations, permutations, or products, or some mix of those. Or it may have to be worked out by looking at it from a series of different perspectives.

Part (a) is straightforward; I would write a product. Please show what you have done there, and then show whatever you are able to do on the others. We can start there in helping you see what to do.
 
I'm even confused with part (a).
I know how to do if password was length 8, and contained a mix of uppercase letters, lowercase letters, and digit, n = 26+26+10 = 62. Number of selections = 8. Total password combination = 62^8.

However, how would I incorporate the restrictions on only 1 uppercase, 5 lowercase and 2 digits allowed?
 
HappiBoi6 said:
I'm even confused with part (a).
I know how to do if password was length 8, and contained a mix of uppercase letters, lowercase letters, and digit, n = 26+26+10 = 62. Number of selections = 8. Total password combination = 62^8.

However, how would I incorporate the restrictions on only 1 uppercase, 5 lowercase and 2 digits allowed?
Click to expand...
What you've done doesn't help; none of the places in the password is free to contain any type of character, so that's an extreme overcount. But you've shown that you know the basic idea.

The problem says,

... the password has length 8, being a mix of 1 uppercase letter (from {A, ..., Z}), 5 lowercase letters (from {a, ..., z}), and 2 digits (from {0, ..., 9}). They do not know in which order these symbols occur in the password.​

So you can form a possible password by first choosing which spaces use each type of character, and then choosing the appropriate character for each.

How many ways can you choose which spaces contain each type? You might think of this as arranging ULLLLLDD in any possible order; or as choosing where to put the one uppercase, then where to put the two digits, then the rest are all lowercase.

Now, how many ways could you choose 1 uppercase, then 5 lowercase, then 2 digits, to go in those spaces?
 
HappiBoi6 said:
I'm even confused with part (a).
I know how to do if password was length 8, and contained a mix of uppercase letters, lowercase letters, and digit, n = 26+26+10 = 62. Number of selections = 8. Total password combination = 62^8.
However, how would I incorporate the restrictions on only 1 uppercase, 5 lowercase and 2 digits allowed?
Click to expand...
Actually part b) has a simple answer compared with part a).
With part b) the answer is \(\displaystyle \binom{26}{1}\cdot\binom{26}{5}\cdot\binom{10}{2}\cdot(8!)\).
Because the symbols are all distinct we can use simple combinations to select which we use.
Example: \(\displaystyle Abztek96\) that one choice can be arranged in \(\displaystyle 8!\) ways.

But for part a) we might select \(\displaystyle Tzzzyy00\) that is one possible example from \(\displaystyle 26\cdot26^5\cdot10^2\) content-wise.
But that one example can be arranged in \(\displaystyle \frac{8!}{3!\cdot(2!)^2}\) ways
What makes part a) so difficult is the fact at the parts can be partitioned in many ways.
 
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