Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In recent years, the state of California issued license plates using a combination of one letter of the alphabet followed by three digits, followed by another three letters of the alphabet. How many different license plates can be issued using this configuration?

Step-by-Step Solution

Answer

There are 175,760,000 different license plates that can be issued using the given configuration, calculated by applying the counting principle: \( 26 * 10 * 10 * 10 * 26 * 26 * 26 \).

Step 1: Count Choices for First Position

The first position can be filled by any of the 26 letters of the alphabet. Therefore, there are 26 choices for the first position.

Step 2: Count Choices for Second, Third, and Fourth Positions

The second, third, and fourth positions can be filled with any of the 10 digits (0-9). Hence, there are 10 choices for each of these positions.

Step 3: Count Choices for Fifth, Sixth, and Seventh Positions

The fifth, sixth, and seventh positions can be filled with any of the 26 letters of the alphabet. Therefore, there are 26 choices for each of these positions.

Step 4: Apply Counting Principle

The counting principle states that we multiply the number of choices available per position to find the total number of combinations. Thus, the total number of different license plates will be: Total combinations = First_position_choices * Second_position_choices * Third_position_choices * Fourth_position_choices * Fifth_position_choices * Sixth_position_choices * Seventh_position_choices

Step 5: Calculate Total Combinations

Substitute the values calculated earlier for the choices for each position: Total combinations = 26 * 10 * 10 * 10 * 26 * 26 * 26

Step 6: Compute the Result

The final multiplication to obtain the total number of different license plates: Total combinations = 175,760,000 Therefore, there are 175,760,000 different license plates that can be issued using the given configuration.

Key Concepts

Counting Principle

The Counting Principle is a fundamental concept in combinatorics. It helps us to determine the total number of possible outcomes when there are multiple steps involved in a process. If each step involves a set number of choices, and these choices are independent of each other, we can find the total combinations by simply multiplying the number of choices for each step.
In the context of the license plate problem, the Counting Principle was used to calculate the total number of possible license plates. There were 26 choices for the first letter, 10 choices for each of the three subsequent digits, and 26 choices for each of the last three letters.
By applying the Counting Principle, we multiplied all these options together:

• 26 (first letter choice),
• 10 (for each of the three digits),
• 26 (for each of the last three letters).

This results in a total of 175,760,000 different possibilities. This principle ensures that all combinations are considered.

Permutations

Permutations are arrangements of a set of items where the order matters. With permutations, every different order of the items counts as a different arrangement. This concept can be particularly useful and important when the sequence of the options affects the outcome.
In the scenario of license plates, if we were only rearranging the letters of the alphabet without repetition, permutations would be a direct application. However, because the position of numbers and letters in a California license plate is fixed, permutations aren't used directly for calculating the total outcomes.
Nonetheless, understanding permutations is crucial when each license plate component could change its sequence. If order was relevant for parts of the setup, permutations calculations would be appropriate to apply.

Combinations

Combinations offer a distinct concept from permutations in combinatorics, where the order does not matter. When picking a combination, you are selecting items in such a way that different orders do not create new combinations.
This is not applicable to the California license plate problem because each position on the plate must be filled with a letter or number in a specific place, and the order of these matters for forming a new license plate.
If we were dealing with a scenario like choosing a committee from a larger group of people where the order of selection doesn't matter, combinations would be used.

• They are calculated by determining how many ways you can choose a subset of items from a larger set without concern for order.

Combinations are calculated using the formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \(n\) is the total number of items to choose from, \(r\) is the number of items to choose, and \(n!\) denotes the factorial of \(n\). Understanding combinations helps us when such non-ordered selections are needed.