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?
The Correct Answer and Explanation is:
To calculate how many license plates can be generated using the format described, let’s break it down:
Problem Understanding
The format consists of 22 letters and 9 digits, and both letters and digits can be repeated. The total number of available choices for each character in the license plate will be based on how many letters and digits there are.
- There are 22 letters to choose from (assuming the alphabet is used, and the number of letters is constant).
- There are 9 digits to choose from (from 0 to 9).
License Plate Format
Let’s assume the license plate has a fixed number of characters where:
- Each character can be one of 22 letters or 9 digits, and we can repeat the choices.
Step 1: Number of Choices per Character
For each position in the license plate, we can either choose a letter or a digit. Therefore:
- For the letters, there are 22 possible choices.
- For the digits, there are 9 possible choices.
Step 2: Number of Possible License Plates
Now, we calculate the total number of possible combinations. If the license plate consists of 6 characters, for example, then:
- For each of the 6 positions, you can either choose from 22 letters or 9 digits, making a total of 22 + 9 = 31 choices for each position.
- Since repetition is allowed, the total number of license plates is the product of the choices for each character.
The total number of license plates NNN is:N=31×31×31×31×31×31N = 31 \times 31 \times 31 \times 31 \times 31 \times 31N=31×31×31×31×31×31
This simplifies to:N=316N = 31^6N=316
Step 3: Calculate 31631^6316
Let’s now compute the value:316=887,503,68131^6 = 887,503,681316=887,503,681
Conclusion
Thus, the total number of license plates that can be generated is 887,503,681. This is assuming that the license plate format has 6 positions and each position can be either a letter (22 choices) or a digit (9 choices), with repetition allowed.
