Question
NY license plates have 3 letters followed by 4 numbers.
a. If the same letter or number can be repeated, how many license plates can be made?
b. If the same letter and number cannot be repeated, how many can be made?
a. If the same letter or number can be repeated, how many license plates can be made?
b. If the same letter and number cannot be repeated, how many can be made?
Answers
GPT-4o mini
To solve the problem of how many license plates can be made in New York with the given format of 3 letters followed by 4 numbers, we will analyze both scenarios: one where repetitions are allowed and another where they are not.
### a. If the same letter or number can be repeated:
1. **Choosing the letters:**
- There are 26 letters in the English alphabet.
- For each of the 3 letter positions, we can choose any of the 26 letters.
- Thus, the total number of ways to select the letters is:
\[
26 \times 26 \times 26 = 26^3
\]
2. **Choosing the numbers:**
- There are 10 digits (0 through 9).
- For each of the 4 number positions, we can choose any of the 10 digits.
- Hence, the total number of ways to select the numbers is:
\[
10 \times 10 \times 10 \times 10 = 10^4
\]
3. **Total license plates:**
- To find the total number of possible license plates, we multiply the number of choices for letters and numbers:
\[
\text{Total} = 26^3 \times 10^4
\]
Calculating the values:
\[
26^3 = 17576 \quad \text{and} \quad 10^4 = 10000
\]
Thus,
\[
\text{Total} = 17576 \times 10000 = 175760000
\]
### b. If the same letter and number cannot be repeated:
1. **Choosing the letters:**
- For the first letter, there are 26 options.
- For the second letter, there are 25 remaining options (since we can't repeat the first letter).
- For the third letter, there are 24 remaining options.
- So, the total number of ways to select the letters is:
\[
26 \times 25 \times 24
\]
2. **Choosing the numbers:**
- For the first number, there are 10 options.
- For the second number, there are 9 remaining options (since we can't repeat the first number).
- For the third number, there are 8 remaining options.
- For the fourth number, there are 7 remaining options.
- So, the total number of ways to select the numbers is:
\[
10 \times 9 \times 8 \times 7
\]
3. **Total license plates:**
- To find the total number of possible license plates, we multiply the number of choices for letters and numbers:
\[
\text{Total} = (26 \times 25 \times 24) \times (10 \times 9 \times 8 \times 7)
\]
Calculating the values:
\[
26 \times 25 \times 24 = 15600
\]
\[
10 \times 9 \times 8 \times 7 = 5040
\]
Thus,
\[
\text{Total} = 15600 \times 5040 = 78744000
\]
### Summary of Results:
- **a.** With repetitions allowed: **175760000**
- **b.** Without repetitions: **78744000**
### a. If the same letter or number can be repeated:
1. **Choosing the letters:**
- There are 26 letters in the English alphabet.
- For each of the 3 letter positions, we can choose any of the 26 letters.
- Thus, the total number of ways to select the letters is:
\[
26 \times 26 \times 26 = 26^3
\]
2. **Choosing the numbers:**
- There are 10 digits (0 through 9).
- For each of the 4 number positions, we can choose any of the 10 digits.
- Hence, the total number of ways to select the numbers is:
\[
10 \times 10 \times 10 \times 10 = 10^4
\]
3. **Total license plates:**
- To find the total number of possible license plates, we multiply the number of choices for letters and numbers:
\[
\text{Total} = 26^3 \times 10^4
\]
Calculating the values:
\[
26^3 = 17576 \quad \text{and} \quad 10^4 = 10000
\]
Thus,
\[
\text{Total} = 17576 \times 10000 = 175760000
\]
### b. If the same letter and number cannot be repeated:
1. **Choosing the letters:**
- For the first letter, there are 26 options.
- For the second letter, there are 25 remaining options (since we can't repeat the first letter).
- For the third letter, there are 24 remaining options.
- So, the total number of ways to select the letters is:
\[
26 \times 25 \times 24
\]
2. **Choosing the numbers:**
- For the first number, there are 10 options.
- For the second number, there are 9 remaining options (since we can't repeat the first number).
- For the third number, there are 8 remaining options.
- For the fourth number, there are 7 remaining options.
- So, the total number of ways to select the numbers is:
\[
10 \times 9 \times 8 \times 7
\]
3. **Total license plates:**
- To find the total number of possible license plates, we multiply the number of choices for letters and numbers:
\[
\text{Total} = (26 \times 25 \times 24) \times (10 \times 9 \times 8 \times 7)
\]
Calculating the values:
\[
26 \times 25 \times 24 = 15600
\]
\[
10 \times 9 \times 8 \times 7 = 5040
\]
Thus,
\[
\text{Total} = 15600 \times 5040 = 78744000
\]
### Summary of Results:
- **a.** With repetitions allowed: **175760000**
- **b.** Without repetitions: **78744000**