Problem 22 - Precalculus with Limits

Question

In a certain state, each automobile license plate number consists of two letters followed by a four-digit number. To avoid confusion between O and zero and between I and one, the letters O and I are not used. How many distinct license plate numbers can be formed in this state?

Step-by-Step Solution

Answer

The total number of distinct license plate numbers that can be formed is \(24 * 24 * 10 * 10 * 10 * 10 = 5,760,000\).

Step 1: Determine the Total Options for the Letters

Since the English Alphabet consists of 26 letters and 'O' and 'I' cannot be used, there are \(26 - 2 = 24\) options for each letter position. Since there are 2 letter positions, and repetition is allowed, there are \(24 * 24\) options for the two letters.

Step 2: Determine the Total Options for the Numbers

There are 4 digit positions and each digit position can contain any number from 0-9, meaning there are 10 options per digit position. Therefore, since repetition is allowed, there are \(10 * 10 * 10 * 10\) options for the four numbers.

Step 3: Determine the Total Distinct License Plates

Since each letter choice is independent from the digit choice, the total number of distinct license plate numbers is the product of the number of letter options and number options. This implies \(24 * 24 * 10 * 10 * 10 * 10\) distinct license plate numbers.

Key Concepts

Permutation

Permutations refer to the various ways in which a set of items can be arranged, where order matters. Imagine you are arranging alphabets to form different letter combinations on a license plate. If you consider two positions for letters and each can take 24 possible characters (excluding 'O' and 'I'), the arrangement or permutation for these two letters would calculate as follows:

• The first letter can be any one of the 24 letters.
• The second letter can also be any of the 24 letters, allowing repetition.

This results in a total of \(24 \times 24\) permutations for the letters. Permutations are particularly useful when sequence is crucial, like in the specific order of a license plate.

Combination

Combinations are similar to permutations, but the order does not matter. In the context of our license plate problem, we are focused more on permutations because order affects outcomes—changing the arrangement creates a different plate number. However, understanding combinations helps when scenarios demand counting configurations where the sequence is irrelevant. Hence, combinations are often used in probability and other fields, although not applicable directly for our license plate arrangement, they pave the way for grasping broader counting problems.

Probability

Probability is the chance of a particular event occurring out of all possible outcomes. When talking about license plates, if you wanted to find the probability of generating a specific license plate from all possible licenses, you would need to consider probability. Since there are \(24 \times 24 \times 10 \times 10 \times 10 \times 10\) possible plates, the probability of picking any one specific plate, say "AB1234", would be calculated as \(\frac{1}{24 \times 24 \times 10^4}\). This tiny fraction highlights how particular combinations become less likely as choices increase. Understanding probability helps you see the likelihood of achieving certain configurations among vast possibilities.

Counting Principle

The counting principle is a fundamental concept in combinatorics that allows you to determine the total number of possible outcomes in a sequence of events. This principle provides a straightforward way to calculate the number of possible license plates. In the given question, the counting principle aids in combining choices for letters and numbers in one process:

• Start with the choices for letters: \(24\) options for each letter position.
• Multiply these by the options for each number (four digits each with \(10\) possibilities).
• Total number of outcomes for the plate becomes \(24 \times 24 \times 10 \times 10 \times 10 \times 10\).

Each step demonstrates how the principle structures counting systematically. It's a practical tool for any scenarios involving multiple stages with independent choices.