A computer system uses passwords that are exactly seven...

A computer system uses passwords that are exactly seven characters and each character is one of the 26 letters (a-z) or 10 integers $(0-9)$. Uppercase letters are not used. (a) How many passwords are possible? (b) If a password consists of exactly six letters and one number, how many passwords are possible? (c) If a password consists of five letters followed by two numbers, how many passwords are possible?

A computer system uses passwords that are exactly seven characters and each character is one of the 26 letters (a-z) or 10 integers $(0-9)$. Uppercase letters are not used.
(a) How many passwords are possible?
(b) If a password consists of exactly six letters and one number, how many passwords are possible?
(c) If a password consists of five letters followed by two numbers, how many passwords are possible?

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Key Concepts

Counting Under Constraints

When additional restrictions or specific patterns (such as a fixed order of letters and numbers) are imposed, counting under constraints involves applying counting principles appropriately to account for limited choices in certain positions while still using the multiplication principle for the overall count.

Permutations with Repetition

This concept applies when the same character or type of character can be used more than once in different positions. The selection in each position is independent and the choices can be repeated, which is key in calculating the total number of password combinations.

Combinatorial Selection (Binomial Coefficients)

When certain positions within a password are restricted to specific types of characters (e.g., exactly one number among letters), the binomial coefficient is used to count the number of ways to choose positions for these characters without considering their order.

Multiplication Principle

This principle is used when the outcome involves a sequence of independent choices. In password counting, each character is chosen independently from a set, and the total number of outcomes is the product of the number of choices available for each character.

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