Final answer:
To find the number of valid passwords of length 19 with no character repeating, use permutations. The calculation involves multiplying the available choices for each position from 36 down to 18, starting with 36 choices and reducing by one for each subsequent character. The result is 36! / (36-19)!, representing all possible arrangements without repetition.
Step-by-step explanation:
Calculating the Number of Valid Passwords
To calculate the number of valid passwords of length 19 where no character repeats and each character can be a digit [0-9] or a lowercase letter [a-z], we can use the principles of permutations. Since there are 10 possible digits and 26 possible lowercase letters, we have a total of 36 distinct characters to choose from for each position in the password. We start by choosing the first character, for which we have 36 possibilities. For the second character, since we can't repeat the first one, we have 35 possibilities, then 34 for the next, and so on until we choose the 19th character, for which there will be 18 possibilities left (36-18=18).
The number of different valid passwords is calculated as:
36 × 35 × 34 × ... × 18 = 36! / (36-19)!
Breaking this down into a step-by-step calculation:
There are 36 possible characters for the first position in the password.
For each subsequent position, there is one less character available (since no character repeats).
Multiply the number of possibilities for each position together to get the total number of valid passwords.
This kind of problem is a common example of a permutation problem where repetition is not allowed, and is used in combinatorial mathematics to determine the number of possible arrangements of a set where order matters.