Question

Postal codes in Canada consist of three letters and three digits. Letters and digits alternate, as in the code R7B 5K1. a) How many different postal codes are possible with this format? b) Do you think Canada will run out of postal codes? Why or why not?

Short answer

a) 17,576,000 possible postal codes. b) Canada could eventually run out of postal codes due to population growth and administrative requirements.

Step-by-step solution

- Calculate Possible Choices for Each Character

Given that there are three letters and three digits in each postal code, we need to determine the number of possible choices for each of these characters. For letters, there are 26 choices (A-Z), and for digits, there are 10 choices (0-9).

- Calculate the Total Number of Possible Postal Codes

Since the characters alternate starting with a letter, the total number of possible postal codes can be calculated by multiplying the number of choices for each character position. This is represented by the formula: i) number of choices: 26 choices for the 1st letter 10 choices for the 1st digit 26 choices for the 2nd letter 10 choices for the 2nd digit 26 choices for the 3rd letter 10 choices for the 3rd digit Therefore, the total number of different postal codes is: \(Total = 26 \times 10 \times 26 \times 10 \times 26 \times 10\)

- Simplify the Calculation

Now, we multiply these numbers: \(Total = 26^3 \times 10^3\) We get: \(Total = 17576 \times 1000 = 17576000\)Therefore, there are 17,576,000 possible postal codes.

- Evaluate If Canada Will Run Out of Postal Codes

Canada has a population of roughly 37 million people. With 17,576,000 possible unique postal codes, it means there is approximately one postal code for every 2 individuals, or nearly half the population.While there is a large number of possible postal codes, it is conceivable that the codes could run out with further population growth or administrative requirements, especially if postal codes are also used for business addresses.

Key concepts

Combinatorics

Combinatorics is an area of mathematics that deals with counting, arrangement, and combination of objects. It particularly helps in solving problems where you need to determine the number of ways to choose items from a set.
Understanding combinatorics is crucial for calculating the total number of possible Canadian postal codes because it involves counting all potential combinations of letters and digits in the format specified.
In our postal code problem, we need to account for the different ways letters and digits can be arranged in a six-character format: three letters and three digits, alternating. The process involves basic principles of multiplication, assuming each slot can be filled independently of the others.

Probability

Probability measures the likelihood of an event occurring. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
In our postal code problem, the probability of creating a specific postal code is determined by the combinatorial calculations of available letters and digits.
The more potential combinations we have, the lower the probability that any single specific combination occurs purely by chance. For example, with 17,576,000 possible combinations, the probability that any particular postal code (like R7B 5K1) is selected randomly is 1 in 17,576,000. This low probability illustrates the vast number of combinations possible, reinforcing the idea that running out of postal codes is unlikely in the immediate term.

Problem-Solving

Effective problem-solving involves identifying the problem, breaking it down into manageable parts, systematically working through these parts, and then combining the solutions.
In our case, the problem is calculating the total number of unique postal codes in Canada and assessing whether they might run out.
We broke down the postal code into individual characters, identified how many options there were for each character (letters and digits), and multiplied the number of choices for each slot. This step-by-step breakdown led us to the final total.
Ultimately, we also evaluated the broader implications by comparing our results with Canada's population, showcasing an essential aspect of problem-solving: interpreting and applying findings to real-world contexts.
Such systematic approaches help in understanding and resolving more complex problems effectively.