Let's solve the problem step-by-step:
### Part 1 of 5:
(a) How many different passwords are possible if each character may be any lowercase letter or digit?
1. Each character of the password can be one of the 26 lowercase letters (`a-z`) or one of the 10 digits (`0-9`).
2. This gives a total of [tex]\( 26 + 10 = 36 \)[/tex] possible options for each character.
3. Since the password is 10 characters long and each character is chosen independently, the total number of different passwords can be calculated using the formula [tex]\(36^{10}\)[/tex].
Using scientific notation with two digits of accuracy after the decimal point:
The possible number of different passwords is approximately [tex]\(3.66 \times 10^{15}\)[/tex].
### Part 2 of 5:
(b) How many different passwords are possible if each character may be any lowercase letter?
1. Each character of the password can be one of the 26 lowercase letters (`a-z`).
2. This gives a total of 26 possible options for each character.
3. Since the password is 10 characters long and each character is chosen independently, the total number of different passwords can be calculated using the formula [tex]\(26^{10}\)[/tex].
Using scientific notation with two digits of accuracy after the decimal point:
The possible number of different passwords is approximately [tex]\(1.41 \times 10^{14}\)[/tex].
Therefore, the answer for Part 2(b) is:
The possible number of different passwords is [tex]\(1.41 \times 10^{14}\)[/tex].
