Let's solve this problem step-by-step using the principles of counting with repetition allowed.
1. Identify Usable Letters and Digits:
- We are given that certain letters and digits are not used. The usable letters total 22 since we exclude I, O, Q, and U from the 26 letters of the alphabet (26 - 4 = 22).
- The usable digits total 6 since we exclude 0, 1, 2, and 3 from the 10 digits (10 - 4 = 6).
2. Determine the License Plate Format:
- Each license plate consists of 4 letters followed by 2 digits. Therefore, a plate looks like LLLLDD, where L represents a letter and D represents a digit.
3. Calculate the Total Number of Possible Plates:
- Since each position on the plate can be any of the usable letters or digits, we use the counting principle where repetition is allowed.
- For the 4 letters (LLLL), each of the 4 positions can be any of the 22 letters. Therefore, the number of combinations for the letters is [tex]\(22 \times 22 \times 22 \times 22 = 22^4\)[/tex].
- For the 2 digits (DD), each of the 2 positions can be any of the 6 digits. Therefore, the number of combinations for the digits is [tex]\(6 \times 6 = 6^2\)[/tex].
4. Combine the Two Calculations:
- The total number of different license plates is the product of the combinations for the letters and the digits.
- This gives us [tex]\(22^4 \times 6^2\)[/tex].
5. Find the Numerical Value:
- We know that [tex]\(22^4\)[/tex] and [tex]\(6^2\)[/tex] can be calculated separately:
- [tex]\(22^4 = 234256\)[/tex]
- [tex]\(6^2 = 36\)[/tex]
- Multiplying these results, the total number of license plates is:
[tex]\[
22^4 \times 6^2 = 234256 \times 36 = 8433216
\][/tex]
Therefore, the total number of possible license plates that can be generated using this format is 8,433,216.
