How many 3-5 character passwords are there if each character is either an
uppercase letter A-Z, a lowercase letter a-z, or a digit 0-9, and where at
least one character of each of the three types is used?
To find the number of passwords, we can use the concept of counting principles.
First, let's determine the number of possibilities for each type of character:
1. Uppercase letters (A-Z): There are 26 possibilities.
2. Lowercase letters (a-z): There are also 26 possibilities.
3. Digits (0-9): There are 10 possibilities.
Now, let's consider the different cases for the length of the password:
Case 1: 3-character password
In this case, we need at least one character from each type.
- The first character can be chosen from any of the three types (26 + 26 + 10 = 62 possibilities).
- The second character can also be chosen from any of the three types (62 possibilities).
- The third character can be chosen from any of the three types as well (62 possibilities).
So, the total number of valid 3-character passwords is 62 * 62 * 62 = 238,328 possibilities.
Case 2: 4-character password
In this case, we still need at least one character from each type.
- The first character can be chosen from any of the three types (62 possibilities).
- The second and third characters can also be chosen from any of the three types (62 possibilities each).
- The fourth character must be from the remaining type (2 possibilities).
So, the total number of valid 4-character passwords is 62 * 62 * 62 * 2 = 8,61,744 possibilities.
Case 3: 5-character password
Similarly, we need at least one character from each type.
- The first character can be chosen from any of the three types (62 possibilities).
- The second, third, and fourth characters can also be chosen from any of the three types (62 possibilities each).
- The fifth character must be from the remaining type (1 possibility).
So, the total number of valid 5-character passwords is 62 * 62 * 62 * 1 = 238,328 possibilities.
Finally, to get the total number of valid passwords, we sum the possibilities from each case:
Total = 238,328 + 8,61,744 + 238,328 = 11,38,400.
Therefore, there are 11,38,400 different 3-5 character passwords that satisfy the given conditions.