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 find the number of 5-digit numbers that can be formed using the digits from the set {1, 2, 3, 4} such that all four digits are included, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to create 5-digit numbers using the digits 1, 2, 3, and 4, ensuring that all four digits are present in each number.

2. **Identifying Cases**:
Since we have 5 positions to fill and only 4 unique digits, one of the digits must repeat. We can consider the digit that repeats as the fifth digit. There are 4 possible cases based on which digit is repeated:
- Case 1: The digit 1 is repeated.
- Case 2: The digit 2 is repeated.
- Case 3: The digit 3 is repeated.
- Case 4: The digit 4 is repeated.

3. **Arranging the Digits**:
For each case, we will have 5 digits to arrange, where one digit appears twice. The arrangement can be calculated using the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \cdot p_2! \cdot \ldots \cdot p_k!} \]
where \(n\) is the total number of items to arrange, and \(p_i\) is the count of each distinct item.

In our case:
- Total digits (n) = 5 (1, 1, 2, 3, 4) for Case 1 (repeating 1).
- The arrangement formula becomes:
\[ \text{Arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]
This is the same for each case since the arrangement of digits will yield the same count.

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

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

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