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 of three letters that reads the same backward and forward.

It can be created 1575 license plated with at least one three-piece palindrome.

Step-by-step explanation:

If a license plates has at least one three-piece palindrome, we have the following cases:

1. The license plate just has a palindrome in the three numbers: The number license plates for this case can be calculated using the rule of multiplication as:

5 * 5 * 1 * 3 * 3 * 2 = 450

1st Number 2nd 3rd 1st Letter 2nd 3rd

For have a palindrome in the numbers, the first and the 3rd number need to be the same. So, we have 5 digits for the first and the 2nd Number and one digit for the 3rd Number.

On the other hand, if the palindrome is just in the numbers, the 3rd letter need to be different from the 1st. So we have 3 options for the 1st and the 2nd letter and 2 options for the 3rd letter.

At the same way, we can calculated the number of license plates for the other cases:

2. The license plate just has a palindrome in the three letters:

5 * 5 * 4 * 3 * 3 * 1 = 900

1st Number 2nd 3rd 1st Letter 2nd 3rd

3. The license plate has a palindrome in the three numbers and in the letters:

5 * 5 * 1 * 3 * 3 * 1 = 225

1st Number 2nd 3rd 1st Letter 2nd 3rd

Then, It can be created 1575 license plates with at least one three-piece palindrome and it is calculated as:

450 + 900 + 225 = 1575