Problem 28
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: Are possible.
Step-by-Step Solution
Verified Answer
There are 1,757,600 possible zip codes in Canada following the given format. This is calculated using the multiplication principle and considering 26 options for each letter position and 10 options for each digit position: \(26^3 * 10^3\) = 1,757,600.
1Step 1: Analyzing the format of the zip code
Since the Canadian zip code format is in the form of L1D2L3D4L5D6, where L represents a letter and D represents a digit, we have 3 positions for letters and 3 positions for digits in the code.
2Step 2: Determine the options for each letter position
Since there are 26 letters in the alphabet, and each letter position can take any letter, we have 26 options for each letter position.
3Step 3: Determine the options for each digit position
Since there are 10 digits (0-9), and each digit position can take any digit, we have 10 options for each digit position.
4Step 4: Calculate the total number of combinations
To calculate the total number of possible zip codes, we need to find the number of ways to arrange the 3 letters and 3 digits.
Using the multiplication principle, we multiply the number of options for each position:
Number of possible Zip Codes = (26 choices for first letter) x (10 choices for first digit) x (26 choices for second letter) x (10 choices for second digit) x (26 choices for third letter) x (10 choices for third digit)
5Step 5: Calculate the result
Now we will calculate the result by multiplying the numbers:
Number of possible Zip Codes = 26 x 10 x 26 x 10 x 26 x 10 = \(26^3 * 10^3\) = 1,757,600
Thus, there are 1,757,600 possible zip codes in Canada following the given format.
Key Concepts
Permutation and CombinationMultiplication PrincipleDiscrete Mathematics
Permutation and Combination
Understanding the basics of permutation and combination is key to solving problems involving the arrangement of different items. In the context of Canadian zip codes, which are composed of a combination of letters and digits, both concepts are applicable.
Permutation refers to the arrangement of items in a specific order, where the sequence is important. In the case of the zip code problem, if the order of the digits and letters were crucial, we would consider permutations. However, since the zip code structure has a set alternating pattern between letters and digits, each position is distinct and pre-determined, which simplifies our problem to one of simple counting rather than permutations.
Combination, on the other hand, involves selecting items from a group where order does not matter. This isn't directly relevant to the zip code problem because the position of each letter and digit is fixed. When considering how many different zip codes can be made, we're actually dealing with the combination of three independent choices for letters and three for digits, within a structured permutation preset by the letter-digit pattern.
Permutation refers to the arrangement of items in a specific order, where the sequence is important. In the case of the zip code problem, if the order of the digits and letters were crucial, we would consider permutations. However, since the zip code structure has a set alternating pattern between letters and digits, each position is distinct and pre-determined, which simplifies our problem to one of simple counting rather than permutations.
Combination, on the other hand, involves selecting items from a group where order does not matter. This isn't directly relevant to the zip code problem because the position of each letter and digit is fixed. When considering how many different zip codes can be made, we're actually dealing with the combination of three independent choices for letters and three for digits, within a structured permutation preset by the letter-digit pattern.
Multiplication Principle
The multiplication principle is an integral part of counting in discrete mathematics. It states that if you have a series of decisions or selections to make, and each decision has independent possible outcomes, the total number of outcomes is found by multiplying the number of possibilities for each decision.
In our scenario of calculating possible Canadian zip codes, each position is independent of the others—choosing a letter for the first position doesn't affect the choices available for the second digit, and so forth. Therefore, we use the multiplication principle to determine the total number of zip codes by multiplying the number of choices for each letter position by the choices for each digit position. The zip code problem is a practical application of this principle, demonstrating not just abstract theory but concrete real-world utility.
In our scenario of calculating possible Canadian zip codes, each position is independent of the others—choosing a letter for the first position doesn't affect the choices available for the second digit, and so forth. Therefore, we use the multiplication principle to determine the total number of zip codes by multiplying the number of choices for each letter position by the choices for each digit position. The zip code problem is a practical application of this principle, demonstrating not just abstract theory but concrete real-world utility.
Discrete Mathematics
Discrete mathematics forms the mathematical foundation of computer science and information theory, among other fields. It deals with discrete elements that use distinct values, rather than continuous mathematics, which deals with objects that can vary smoothly. Discrete mathematical structures include sets, graphs, and propositional logic.
In terms of counting and probability, which are parts of discrete mathematics, we often rely on discrete structures like integers and logical statements to solve problems. The Canadian zip code scenario epitomizes a discrete problem—it requires counting a finite number of distinct configurations of letters and digits, which aligns perfectly with the discrete nature of the problem. By recognizing the structured, discrete pattern of Canadian zip codes, we can efficiently apply mathematical concepts like the multiplication principle to find a solution.
In terms of counting and probability, which are parts of discrete mathematics, we often rely on discrete structures like integers and logical statements to solve problems. The Canadian zip code scenario epitomizes a discrete problem—it requires counting a finite number of distinct configurations of letters and digits, which aligns perfectly with the discrete nature of the problem. By recognizing the structured, discrete pattern of Canadian zip codes, we can efficiently apply mathematical concepts like the multiplication principle to find a solution.
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