Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 22
How many possible license plates can be manufactured if a license plate consists of three letters followed by three digits and (a) [BB] the digits must be distinct; the letters can be arbitrary? (b) the letters must be distinct; the digits can be arbitrary? (c) the digits and the letters must be distinct?
Short Answer
Expert verified
(a) \(26^3 \times 720\), (b) \(15600 \times 1000\), (c) \(15600 \times 720\)
Step by step solution
01
Understanding the Problem
We are tasked with finding how many different combinations of license plates can be made given different restrictions on letters and digits.
02
Step 1a: Calculating Letter Combinations (No restrictions)
For a license plate, any of the three letters can be any of the 26 letters from A to Z. Thus, there are \(26 \times 26 \times 26 = 26^3\) ways to choose the letters.
03
Step 1b: Calculating Digit Combinations (Digits must be distinct)
For the digits, each must be unique. Choose 3 digits from 10 options (0-9). There are 10 options for the first digit, 9 for the second, and 8 for the third. So there are \(10 \times 9 \times 8\) ways to choose the digits.
04
Step 1c: Multiplying Combinations for Part (a)
The possible distinct license plates with letter combination \(26^3\) and distinct digit choices \(10 \times 9 \times 8\) are \(26^3 \times 10 \times 9 \times 8\).
05
Step 2a: Calculating Distinct Letter Combinations (Letters must be distinct)
For distinct letters, the first letter has 26 options, the second 25 (must be different from the first), and the third 24. Thus, there are \(26 \times 25 \times 24\) ways to select the letters.
06
Step 2b: Calculating Digit Combinations (No restrictions)
Here we can choose any digit for each place since repetitions are allowed, resulting in \(10 \times 10 \times 10 = 10^3\) digit combinations.
07
Step 2c: Multiplying Combinations for Part (b)
The possible license plates with distinct letters and repeated digit possibilities are \(26 \times 25 \times 24 \times 10^3\).
08
Step 3a: Calculating Distinct Letter Combinations (Letters must be distinct)
Reuse the calculation from step 2a: \(26 \times 25 \times 24\) for distinct letters.
09
Step 3b: Calculating Distinct Digit Combinations (Digits must be distinct)
Reuse the calculation from step 1b: \(10 \times 9 \times 8\) for distinct digits.
10
Step 3c: Multiplying Combinations for Part (c)
The total number of plates having both distinct letters and digits is \(26 \times 25 \times 24 \times 10 \times 9 \times 8\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
License Plate Combinations
License plate combinations involve creating unique identifiers for vehicles using a mix of letters and numbers. These combinations can have restrictions based on how letters and numbers are used. Let's break it down:
- Firstly, we consider the potential letters and numbers for the license plates. Typically, each spot for a letter can be filled with one of 26 possibilities, representing each letter of the alphabet from A to Z.
- Similarly, each spot for a number can be filled with one of 10 possibilities, numbers ranging from 0 to 9.
- The problem becomes trickier when there are rules or restrictions, such as distinct digits or distinct letters, which affect the total number of possible combinations.
Distinct Elements
When dealing with distinct elements, we emphasize that each element (whether a letter or number) must be unique within its specified set, preventing repetition. This requirement can drastically change the number of combinations:
- When digits must be distinct, it means that once a digit has been used in one position, it cannot appear in another position. For example, if `8` is used as the first digit, it cannot be reused as the second or third digit.
- Similarly, if letters must be distinct, once a letter is placed in one of the letter slots, it cannot appear again. If `A` is chosen as the first letter, the remaining choices for the second and third letters exclude `A`.
Permutations
Permutations are arrangements of a set of elements where order matters. It's a key concept when calculating combinations, especially when elements must be distinct:
- For the digit section of a license plate where digits are distinct, permutations help determine how these digits can be uniquely arranged. We have a starting pool of 10 digits, and for each selection, the available options decrease as we pick successive digits.
- Similarly, for the letters, using permutations means arranging the letters without repetition. The first letter has all 26 options, but with each choice, the options diminish by one.
Factorials
Factorials are mathematical tools that help us calculate permutations and combinations. They are especially useful when elements need to be arranged in every possible order:
- The factorial of a number, denoted as \( n! \), is the product of all positive integers up to that number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
- Factorials come into play when arranging a set of elements, often used to calculate the number of ways to arrange `n` distinct elements, as permutations require.
- For our license plate problem, understanding that arranging `n` distinct numbers or letters among themselves involves using factorials helps break down calculation steps, ensuring accuracy.
