Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

License plate numbers In a certain state, automobile license plates start with one letter of the alphabet, followed by five digits \((0,1,2, \ldots, 9)\). Find how many different license plates are possible if (a) the first digit following the letter cannot be 0 (b) the first letter cannot be \(\mathrm{O}\) or \(\mathrm{I}\) and the first digit cannot be 0

Step-by-Step Solution

Answer

(a) 234,000 plates, (b) 216,000 plates.

Step 1: Count the possibilities for the letter in case (a)

For part (a), any letter of the alphabet is allowed, providing 26 options since there are 26 letters in total.

Step 2: Determine possibilities for the first digit in case (a)

The first digit cannot be 0, which leaves us with the choices 1 through 9. This provides 9 possible choices.

Step 3: Determine possibilities for the remaining four digits in case (a)

Each of the remaining four digits has no restriction and can be any digit from 0 to 9. This means there are 10 possible choices for each of the 4 positions.

Step 4: Calculate the total number of license plates for case (a)

Multiply the possibilities for each position of the license plate. So, it is \(26 \times 9 \times 10^4 = 234,000\) possible license plates.

Step 5: Refine the letter choices for case (b)

In case (b), the letters O and I are not allowed. Therefore, there are 24 possible letter choices, excluding O and I.

Step 6: Count digit possibilities for case (b)

Similar to step 2, the first digit still cannot be 0. This results in 9 choices for the first digit.

Step 7: Count possibilities for remaining digits in case (b)

As in step 3, the remaining four digits each have 10 possibilities. Hence, there are 10 options per position for these digits.

Step 8: Calculate the total number of license plates for case (b)

Multiply the possibilities for each position of the license plate, considering the restrictions: \(24 \times 9 \times 10^4 = 216,000\) possible plates.

Key Concepts

Probability in Combinatorics

Probability in combinatorics focuses on counting the number of possible outcomes and relating these to the desired outcome. In the context of license plates, we are interested in the likelihood of constructing a valid plate within given restrictions. For example, if each digit on a license plate can be from 0 to 9, but only non-zero digits are allowed first, the total probability must account for these constraints. This means narrowing down the selection and calculating the chances based on limited options. Understanding these probability questions relies on our ability to count possible arrangements effectively.

Number Theory and Digits

Number theory often deals with the properties of numbers, particularly integers. In combinatorics, it helps us understand the use of digits and the numerical restrictions involved. When assigning numbers to a license plate, you cannot use a 0 in the first digit as it affects the mathematical representation and readability of the plate. Therefore, instead of all ten digits being options (0-9), only digits 1 through 9 are available for the first position. This example highlights how number theory can dictate rules in everyday contexts, demanding valid, understandable, and logical sequences of numbers.

Permutations and Combinations

Permutations and combinations are mathematical concepts that explore different ways to arrange or select items. In this scenario, we have to arrange a letter and five digits in a specific order for a license plate. Specifically, permutations are concerned with how the positioning of items yields different outcomes. For example, arranging letters and numbers gives us unique license plate variations based on their sequence. Calculating combinations involves deciding how many different selections can be made when order does not matter, but here, the order is essential, hence the need for permutations. By understanding these principles, we can determine exactly how many possible plates exist given the restrictions provided.

Navigating Mathematical Restrictions

Mathematical restrictions often guide how we must structure our solutions to account for specific rules. For instance, in this exercise, only 24 of the alphabet's 26 letters are permitted due to the prohibition of 'O' and 'I', which resemble the digits '0' and '1'. Similarly, the first numerical digit cannot be 0, leaving only nine options (1-9). Such constraints simplify complex real-world issues and ensure practical, reliable outcomes in various applications. By understanding these limitations, we can calculate the precise number of viable registration plate combinations, adapting to any additional constraints provided.