In a telephone system four different letter `P,R, S, T` and the four digits `3, 5, 7, 8` are used. Find the maximum number of "telephone numbers" the system can have if each consists of a letter followed by a four-digit number in which the digit may be repeated.

Options

A. 1024
B. 2048
C. `4^(5)`
D. `5^(4)`

Answer

To find the maximum number of telephone numbers that can be formed using the letters `P, R, S, T` and the digits `3, 5, 7, 8`, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
Each telephone number consists of one letter followed by a four-digit number. The letters available are `P, R, S, T`, and the digits available are `3, 5, 7, 8`.

2. **Count the Choices for the Letter**:
There are 4 different letters (P, R, S, T). Therefore, there are 4 choices for the first position (the letter).

3. **Count the Choices for Each Digit**:
The four-digit number can have digits that may be repeated. For each of the four digit positions, we can choose any of the 4 digits (3, 5, 7, 8).

- For the first digit: 4 choices
- For the second digit: 4 choices
- For the third digit: 4 choices
- For the fourth digit: 4 choices

4. **Calculate the Total Choices for the Digits**:
Since the digits can be repeated, the total number of combinations for the four-digit number is:
\[ 4 \times 4 \times 4 \times 4 = 4^4 \]

5. **Combine the Choices**:
The total number of telephone numbers can be calculated by multiplying the number of choices for the letter by the number of choices for the four-digit number:
\[ \text{Total Telephone Numbers} = \text{Choices for Letter} \times \text{Choices for Digits} \]
\[ = 4 \times 4^4 \]

6. **Simplify the Expression**:
We can simplify \(4 \times 4^4\) as follows:
\[ 4 \times 4^4 = 4^1 \times 4^4 = 4^{1+4} = 4^5 \]

7. **Calculate \(4^5\)**:
Now we calculate \(4^5\):
\[ 4^5 = 1024 \]

### Final Answer:
Thus, the maximum number of telephone numbers that can be formed is **1024**.