Typically, radio stations are identified by four "call...

Typically, radio stations are identified by four "call letters." Radio stations east of the Mississippi River have call letters that start with the letter $\mathrm{W},$ and radio stations west of the Mississippi River have call letters that start with the letter $\mathrm{K}.$ (a) Find the number of different sets of radio station call letters that are possible in the United States. (b) Find the number of different sets of radio station call letters that are possible when the call letters must include a $\mathrm{Q}.$

Typically, radio stations are identified by four "call letters." Radio stations east of the Mississippi River have call letters that start with the letter $\mathrm{W},$ and radio stations west of the Mississippi River have call letters that start with the letter $\mathrm{K}.$
(a) Find the number of different sets of radio station call letters that are possible in the United States.
(b) Find the number of different sets of radio station call letters that are possible when the call letters must include a $\mathrm{Q}.$

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Key Concepts

Multiplication Principle

The multiplication principle is a fundamental counting concept that states if there are 'm' ways to do one thing and 'n' ways to do another, then there are m × n ways of performing both actions in sequence. This concept is used to determine the total number of possible outcomes when making sequential independent choices, such as selecting each letter of a radio station's call sign from a set of options.

Case Analysis

Case analysis involves breaking down a counting problem into distinct scenarios or cases that are easier to manage separately. In problems where constraints differ based on conditions—like the first letter being determined by geographical location—each case can be counted individually and then added together to obtain the overall total.

Complement Principle

The complement principle is a useful tool for counting situations where it may be easier to count the total number of possibilities and then subtract those that do not meet a particular criterion. This technique simplifies solving problems with restrictions, such as ensuring at least one letter is a specific character, by finding the total number of outcomes and deducting the count of outcomes that violate the condition.

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