Consider all the 5 digit numbers where each of the digits is chosen from the set { 1, 2, 3, 4} . Then the number of numbers, which contain all the four digits is :

Options

A. 240
B. 244
C. 586
D. 781

Answer

To solve the problem of finding the number of 5-digit numbers that can be formed using the digits from the set {1, 2, 3, 4} while ensuring that all four digits are included, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to create a 5-digit number using the digits 1, 2, 3, and 4, ensuring that each digit appears at least once.

2. **Identifying Cases**:
Since we have to use all four digits, one of the digits must appear twice. We can break this down into different cases based on which digit appears twice.

- **Case 1**: The digit '1' appears twice, and '2', '3', '4' appear once each.
- **Case 2**: The digit '2' appears twice, and '1', '3', '4' appear once each.
- **Case 3**: The digit '3' appears twice, and '1', '2', '4' appear once each.
- **Case 4**: The digit '4' appears twice, and '1', '2', '3' appear once each.

3. **Calculating Arrangements for Each Case**:
For each case, we need to calculate the number of distinct arrangements of the digits.

The formula for the number of arrangements of n objects where one object is repeated \( r \) times is given by:
\[ \frac{n!}{r!} \]
In our case, we have 5 digits in total, where one digit is repeated twice.

Therefore, for each case, the number of arrangements is:
\[ \frac{5!}{2!} \]

4. **Calculating Factorials**:
- \( 5! = 120 \)
- \( 2! = 2 \)

Thus, the number of arrangements for each case is:
\[ \frac{120}{2} = 60 \]

5. **Total Arrangements**:
Since there are 4 cases (one for each digit being repeated), we multiply the arrangements from one case by the number of cases:
\[ \text{Total} = 4 \times 60 = 240 \]

### Final Answer:
The total number of 5-digit numbers that can be formed using the digits {1, 2, 3, 4} such that all four digits are included is **240**.