Problem 20
Question
A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that: Have the property that both words and numbers are palindromes.
Step-by-Step Solution
Verified Answer
The total number of palindromic license plates is 60,840.
1Step 1: Palindromic 3-Letter Words
In a palindromic 3-letter word, the first letter must match the third letter. So, to form such a word, we only need to choose the first and third letter (since they will be the same) and the middle letter. There are 26 letters in the English alphabet.
There are 26 choices for the first and third letter and 26 choices for the middle letter, so there are a total of 26*26 = 676 palindromic 3-letter words.
2Step 2: Palindromic 3-Digit Numbers
Like the 3-letter words, a palindromic 3-digit number requires that the first digit matches the last digit. So, we only need to choose the first and third digit (since they will be the same) and the middle digit. There are 10 digits (0-9) but the first digit cannot be 0.
There are 9 choices for the first and third digit (1 to 9) and 10 choices for the middle digit (0 to 9), so there are a total of 9*10 = 90 palindromic 3-digit numbers.
3Step 3: Combine the Number of Palindromic Words and Digits
Now, we need to combine the number of palindromic 3-letter words (676) and the number of palindromic 3-digit numbers (90). Since the words and numbers are independent, we multiply the number of palindromic words by the number of palindromic numbers.
So, the total number of palindromic license plates is: 676 * 90 = 60840.
Key Concepts
PalindromeLicense Plate ProblemCombinatorial CountingPermutations
Palindrome
A palindrome is a sequence that reads the same forwards and backwards. This concept is at the heart of our license plate problem. For a 3-letter word to be palindromic, the first and the third letters must be identical. The middle letter can be any letter from the alphabet.
Therefore, if we want to create a palindromic sequence with three letters, we select the first letter (which is also the third) from the 26 options available in the English alphabet. Then, we choose the middle letter, which has the same 26 options. This results in a total of 26 x 26 = 676 different palindromic words.
With numbers, the concept remains similar. A 3-digit number is palindromic if the first and third digits are the same. Here, the twist is that the first digit cannot be zero, otherwise, it wouldn't be a 3-digit number. Thus, we have 9 options (1 through 9) for the first and third digits, and 10 options (0 through 9) for the middle digit, giving us 9 x 10 = 90 palindromic numbers.
Therefore, if we want to create a palindromic sequence with three letters, we select the first letter (which is also the third) from the 26 options available in the English alphabet. Then, we choose the middle letter, which has the same 26 options. This results in a total of 26 x 26 = 676 different palindromic words.
With numbers, the concept remains similar. A 3-digit number is palindromic if the first and third digits are the same. Here, the twist is that the first digit cannot be zero, otherwise, it wouldn't be a 3-digit number. Thus, we have 9 options (1 through 9) for the first and third digits, and 10 options (0 through 9) for the middle digit, giving us 9 x 10 = 90 palindromic numbers.
License Plate Problem
The license plate problem is a nifty challenge that explores how to form unique combinations using letters and numbers. In the context of this problem, we’re specifically concentrating on license plates that resemble palindromes. These are structured as three letters followed by three numbers.
The task is to determine how many such license plates can be constructed when both the alphabet portion and the numerical portion each form a palindrome. Understanding each section's requirements separately and then combining the results is key to solving this problem efficiently.
By calculating the number of possible palindromic 3-letter words and palindromic 3-digit numbers, the ultimate goal is to find how they can be paired together in license plates. Once you have the number of palindromic words and numbers, you multiply these to find the total number of palindromic license plates.
The task is to determine how many such license plates can be constructed when both the alphabet portion and the numerical portion each form a palindrome. Understanding each section's requirements separately and then combining the results is key to solving this problem efficiently.
By calculating the number of possible palindromic 3-letter words and palindromic 3-digit numbers, the ultimate goal is to find how they can be paired together in license plates. Once you have the number of palindromic words and numbers, you multiply these to find the total number of palindromic license plates.
Combinatorial Counting
Combinatorial counting is the mathematical technique used to figure out how many different combinations or arrangements are possible. In problems like our license plate one, it assists in breaking down complex counting situations into manageable parts.
Here, we use combinatorial counting to handle two distinct computations:
After finding the number of possible palindromic words and numbers separately, combinatorial counting guides us to multiply these results, reflecting the principle of independence in these choices.
Here, we use combinatorial counting to handle two distinct computations:
- The calculation of palindromic 3-letter words.
- The calculation of palindromic 3-digit numbers.
After finding the number of possible palindromic words and numbers separately, combinatorial counting guides us to multiply these results, reflecting the principle of independence in these choices.
Permutations
Permutations refer to the various ways we can arrange a set of items. In this license plate problem, though, permutations take on a special form due to the palindrome constraint.
Typically, a permutation of a sequence would mean any ordering of those items. However, in a palindromic arrangement, the regular constraints of order are relieved somewhat by the symmetrical requirements.
Since we need only select the first and third letter or digit (which must be the same) and fill in the middle independently, fewer permutations are required to create a valid sequence. This reduces the complexity of selecting arrangements simplified into a smaller problem, guided by palindromic requirements.
Overall, this simplifies what could have been a massive number of possible permutations into more feasible, well-counted options, ensuring we stay focused on the specific criteria of palindrome formations.
Typically, a permutation of a sequence would mean any ordering of those items. However, in a palindromic arrangement, the regular constraints of order are relieved somewhat by the symmetrical requirements.
Since we need only select the first and third letter or digit (which must be the same) and fill in the middle independently, fewer permutations are required to create a valid sequence. This reduces the complexity of selecting arrangements simplified into a smaller problem, guided by palindromic requirements.
Overall, this simplifies what could have been a massive number of possible permutations into more feasible, well-counted options, ensuring we stay focused on the specific criteria of palindrome formations.
Other exercises in this chapter
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