User Names How many user names are possible in Problem 41 if the last character must be a digit?

Verified step by step guidance

1. Identify the total number of characters allowed for the user names and the constraints given in Problem 41. Typically, this involves knowing the length of the user name and the character set allowed for each position.
2. Determine the number of choices for each position in the user name except the last character. For example, if letters and digits are allowed, count how many options are available for each position.
3. Since the last character must be a digit, specify the number of choices for the last character. Usually, digits range from 0 to 9, so there are 10 possible choices for the last character.
4. Calculate the total number of possible user names by multiplying the number of choices for each position. This is done by taking the product of the number of options for each character position, with the last position fixed to digits only.
5. Express the total number of user names as a product: if the user name length is \( n \), and the number of choices for each of the first \( n-1 \) characters is \( c \), then the total number of user names is \( c^{n-1} \times 10 \).

Key Concepts

Counting Principle

The counting principle helps determine the total number of possible outcomes by multiplying the number of choices for each position. When forming usernames, each character position can have a set number of options, and the total possibilities are the product of these options.

Character Set Restrictions

Character set restrictions limit which characters can appear in certain positions. In this problem, the last character must be a digit, reducing the choices for that position to 10 (digits 0-9), while other positions may have different allowed characters.

Permutations with Constraints

Permutations with constraints involve counting arrangements where some positions have specific requirements. Here, the constraint on the last character affects the total number of valid usernames, requiring careful consideration of allowed characters per position.