Hard math problem?

Question

The license plates on all karts in MarioLand contain three numbers followed by three letters. If you know that the MarioLand number system has 5 digits and the letter system has 3 letters, how many license plates can be created that have at least one three-piece palindrome? A three-piece palindrome is a string of three numbers that reads the same backward and forward, or a string or three letters that reads the same backward and forward.

Answer 1

First, let's think about how many three-piece palindromes we can make with five numbers (like 111 or 121). I can choose any of the five numbers to be the first digit and any of the five numbers to be the second digit. But if I want a three-piece palindrome, the third digit has to be the same as the first digit, so I have no choice there. That means there are 5×5×1 = 25 possible three-piece palindromes for the numbers portion. If you want to figure out how many possible numbers portions there are total, you'll be able to choose any of the five numbers for any of the three spaces for a total of 5×5×5 = 125 possible numbers portions.

 

We can do the same thing for the letters: I can choose any of the three letters to be the first one and any of the three letters to be the second one, but then the third letter has to be the same as the first. So, there are 3×3×1 = 9 possible three-piece palindromes for the letters portion. If you want to figure out how many possible letters portions there are total, you'll be able to choose any of the three numbers for any of the three spaces for a total of 3×3×3 = 27 possible letters portions.

 

(In case you need some help visualizing this, if the letters were a, b, and c, the nine possible palindromes would be aaa, aba, aca, bbb, bab, bcb, ccc, cac, and cbc.)

 

 

Now, let's figure out how many license plates have three-piece palindromes. Each of the 25 numbers-portion three-piece palindromes can be matched with each of the 27 total letters portions for a total of 25×27 = 675 license plates with numbers-portion three-piece palindromes. Each of the 9 letters-portion three-piece palindromes can be matched with each of the 125 total numbers portions for a total of 9×125 = 1125 license plates with letters-portion three-piece palindromes.

 

That's a grand total of 675 + 1125 = 1800 license plates with three-piece palindromes. But we double-counted some plates! Anything with both a numbers-portion palindrome and a letters-portion palindrome (like 121 aba) was counted two times, once with the numbers-portion three-piece palindromes and once with the letters-portion three-piece palindromes. So, we need to figure out how many license plates there are with double three-piece palindromes and subtract them from the set. There are 25 numbers-portion three-piece palindromes, each of which could be matched with each of the 9 letters-portion three-piece palindromes to create 25×9 = 225 license plates with double three-piece palindromes.

 

 

That leaves us with a final total of 1800 - 225 = 1575 license plates with at least one three-piece palindrome.