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?

Answer 1

Answer:

1575 license plates with at least one three piece palindrome

Step-by-step explanation:

Fon know the number of license plates that have at least one three piece palindrome is necessary to identify the following cases:

• There is palindrome in the 3 numbers and not in the 3 letters of the license plate: A three piece palindrome happens when the first and the last number is the same. So we can calculate the number of ways in which this can happen using the rule of multiplication as:

__5__      _*__5___* __1___* __3___   *   __3___*    __2___  =  450

1st number    2nd         3rd        1st letter        2nd            3rd

The MarioLand number system has 5 digits so the are 5 options for the first and second number, then if we want a palindrome, the 3rd number has to be the same of the first so the 3rd number just have one option.

Additionally, the MarioLand letter system has 3 letters, so we have 3 options for first and second number but we don't want a palindrome in the letters, so the 3rd letter need to be a number different of the first, that means that it has 2 options.

At the same way we calculate the number of possibilities in the following cases:

• There is palindrome in the 3 letters and not in the 3 numbers of the license plate: There are 900 different ways that apply for this case, this is calculate as:

__5__      _*__5___* __4___* __3___   *   __3___*    __1___  =  900

1st number    2nd         3rd        1st letter        2nd            3rd

• There is palindrome in the 3 letters and in the 3 numbers of the license plate: There are 225 different ways that apply for this case, this is calculate as:

__5__      _*__5___* __1___* __3___   *   __3___*    __1___  = 225

1st number    2nd         3rd        1st letter        2nd            3rd

Finally the total number of license plate that can be created with at least one three piece palindrome is the sum of the possibilities of the three cases:

450+900+225=1575