Question
Standard automobile license plates in a country display 1 numbers, followed by 3 letters, followed by 3 numbers. How many different standard plates are possible in this system? (Assume repetitions of letters and numbers are allowed.) Your answer is:
Ask by Byrd Pierce.
Mar 30,2025 19:39
Upstudy AI Solution
Tutor-Verified Answer
Answer
There are 175,760,000 different standard license plates possible.
Solution
**Step 1:**
For the first character, which is a number, there are \(10\) possibilities (digits \(0\) through \(9\)).
**Step 2:**
For the next 3 characters, which are letters, there are \(26\) possibilities for each letter. Since repetitions are allowed, the total number of possibilities for the 3 letters is
\[
26 \times 26 \times 26 = 26^3.
\]
**Step 3:**
For the last 3 characters, which are numbers, there are \(10\) possibilities for each number. Therefore, the total number of possibilities for these 3 numbers is
\[
10 \times 10 \times 10 = 10^3.
\]
**Step 4:**
Multiply the number of possibilities for each part to find the total number of different standard plates:
\[
10 \times 26^3 \times 10^3 = 10^4 \times 26^3.
\]
**Step 5:**
Calculate the value of \(26^3\):
\[
26^3 = 26 \times 26 \times 26 = 17576.
\]
**Step 6:**
Then, compute the final total:
\[
10^4 \times 17576 = 10000 \times 17576 = 175760000.
\]
**Answer:**
\(175760000\)
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Simplify this solution Mind Expander
To calculate the total number of different standard automobile license plates possible, we first break down the components:
1. For the first digit, there are 10 possible numbers (0-9).
2. For each of the three letters, assuming there are 26 letters in the alphabet, there are 26 options per letter.
3. For the last three numbers, it again offers 10 options each.
The total combinations would be calculated as follows:
(10 options for the first number) * (26 options for the first letter) * (26 options for the second letter) * (26 options for the third letter) * (10 options for the first number) * (10 options for the second number) * (10 options for the third number).
Mathematically, it is:
10 × (26^3) × (10^3) = 10 × 17,576 × 1,000 = 175,760,000.
So, the total number of different standard plates possible is 175,760,000.
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