Questions: Suppose each license plate in a certain state has one digit, followed by four letters, followed by one digit. The letters F, O, S, and X and the digit 0 are not used. So, there are 22 letters and 9 digits that are used. Assume that the letters and digits can be repeated. How many license plates can be generated using this format?

Solution Steps

To determine the number of possible license plates, we need to calculate the total number of combinations for each part of the license plate format. The format is one digit, followed by four letters, followed by one digit. Given that there are 9 possible digits and 22 possible letters, we can use the multiplication principle to find the total number of combinations.

The license plate format includes one digit at the beginning and one digit at the end. Since the digits used are from 1 to 9 (excluding 0), the number of possible digits is: \[ \text{num\_digits} = 9 \]

The license plate format includes four letters in the middle. The letters available are from the English alphabet, excluding \( F, O, S, \) and \( X \). Therefore, the number of possible letters is: \[ \text{num\_letters} = 26 - 4 = 22 \]

Using the multiplication principle, the total number of license plates can be calculated as follows: \[ \text{total\_plates} = \text{num\_digits} \times (\text{num\_letters})^4 \times \text{num\_digits} \] Substituting the values: \[ \text{total\_plates} = 9 \times (22^4) \times 9 \] Calculating \( 22^4 \): \[ 22^4 = 234256 \] Thus, the total number of license plates is: \[ \text{total\_plates} = 9 \times 234256 \times 9 = 18974736 \]

Final Answer