Questions: License plates in a particular state display 3 letters followed by 2 numbers. How many different license plates can be manufactured for this state? There are different license plates that can be manufactured for this state. (Simplify your answer. Type an integer or a fraction.)

Solution Steps

To calculate the total possibilities for the letter part of the license plate, we use the formula $N_{letters}^n$, where $N_{letters}$ is the total number of possible letters and $n$ is the number of positions for letters. Given $N_{letters} = 26$ and $n = 3$, the total possibilities for the letter part is $N_{letters}^n = 26^3 = 17576$.

To calculate the total possibilities for the number part of the license plate, we use the formula $N_{numbers}^m$, where $N_{numbers}$ is the total number of possible numbers for each digit position and $m$ is the number of positions for numbers. Given $N_{numbers} = 10$ and $m = 2$, the total possibilities for the number part is $N_{numbers}^m = 10^2 = 100$.

To find the total number of different license plates that can be manufactured, we multiply the total possibilities for the letter part by the total possibilities for the number part. This gives us $N_{letters}^n \times N_{numbers}^m = 17576 \times 100 = 1757600$.

Final Answer: The total number of different license plates that can be manufactured is 1757600.