Problem 26 - Discrete Mathematics with Applications

Question

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1B2C3. Find the number of zip codes that: Begin with A and end in 3.

Step-by-Step Solution

Answer

There are 250 zip codes that begin with 'A' and end in '3'.

Step 1: Identifying the possibilities for each position

First, let's look at the possibilities for each position within the zip code. Since L1 is always 'A', we have 1 possibility for that position. L2 can be any letter except 'A', so we have 25 possibilities (there are 26 letters in the alphabet, minus the letter 'A'). For D2, there are 10 possibilities since it can be any digit from 0 to 9. Finally, D3 will always be '3', so we have 1 possibility for that position.

Step 2: Calculating the total number of zip codes

Now, let's find the number of possible zip codes with the pattern L1D2L3. Since there is only 1 possibility for L1 and D3, and we have 25 possibilities for L2 and 10 possibilities for D2, we can calculate the number of possible zip codes using the following formula: \(Total\ number\ of\ zip\ codes = Possibilities\ for\ L1 * Possibilities\ for\ D2 * Possibilities\ for\ L2 * Possibilities\ for\ D3\)

Step 3: Plugging in the values and calculating the result

Now, let's plug in the values for each position and calculate the result: \(Total\ number\ of\ zip\ codes = 1 * 10 * 25 * 1 = 250\) So, there are 250 zip codes that begin with 'A' and end in '3'.

Key Concepts

Factorial

Factorials play a crucial role in combinatorics, especially when preparing to understand permutations and combinations.
Factorials are the product of all positive integers up to a specified number. For example, the factorial of 5, written as 5!, is equal to 5 × 4 × 3 × 2 × 1.

• This means 5! = 120.
• It's a way of counting how many ways items can be arranged.

Factorials are particularly useful when you're dealing with permutations, such as arranging items in a specific order. It's important to note that 0! equals 1 by definition, even though it might seem counterintuitive. This rule helps maintain consistency in mathematical calculations involving factorials.
Factorials are a building block for more complex combinatorial calculations, making them essential for anyone diving deeper into the subject.

Counting Principle

The Counting Principle, often referred to as the Fundamental Counting Principle, is a key idea in combinatorics. It helps us calculate the number of ways multiple events can occur. Simply put, if you have two events, and the first event can occur in "m" ways and the second event can happen in "n" ways, then both events together can occur in "m × n" ways.

• This principle can be extended to more than two events.
• It allows for quick multiplication of the possibilities for each part or step in a process.

In the context of a zip code, once you know the number of possibilities for each letter and number, you multiply these together to find the total number of combinations. In the given example, the zip code starts with a letter "A" and ends with "3." By applying the Counting Principle, you multiply the possibilities for each position to find how many such zip codes exist.

Permutations

Permutations refer to the different ways in which a set of items can be arranged or ordered.
Unlike combinations, order matters in permutations.
For permutations of a given set, we use factorials to calculate the total number of arrangements. For example, the number of permutations of 3 items is 3!, which calculates to 6.

• This concept differs from combinations, where order doesn’t matter.
• It's important to know whether you're asked to calculate permutations or combinations.

When dealing with zip codes, permutations help determine how letters and numbers can be sequenced. Though the specific exercise here doesn't need complex permutation calculations because some positions are fixed, understanding permutations will be crucial in more complex scenarios where the order is critical.