Using the digits 1,2,3,4 and 5 only once, how many numbers greater than 41000 can be formed ? (a) 41 (b) 48 (c) 50 (d) 55

Options

A. 41
B. 48
C. 50
D. 55

Answer

To determine how many numbers greater than 41000 can be formed using the digits 1, 2, 3, 4, and 5 exactly once, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Valid Starting Digits**:
To form a number greater than 41000, the first digit must be either 4 or 5.

2. **Case 1: First Digit is 4**:
- If the first digit is 4, the remaining digits to choose from are 1, 2, 3, and 5.
- We can arrange these 4 digits in any order. The number of arrangements of 4 digits is given by \(4!\) (4 factorial).
- Calculation:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
- Therefore, there are 24 numbers that can be formed starting with the digit 4.

3. **Case 2: First Digit is 5**:
- If the first digit is 5, the remaining digits to choose from are 1, 2, 3, and 4.
- Similarly, we can arrange these 4 digits in any order, which is also \(4!\).
- Calculation:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
- Therefore, there are also 24 numbers that can be formed starting with the digit 5.

4. **Total Count of Numbers Greater than 41000**:
- To find the total number of valid numbers, we add the counts from both cases:
\[ 24 \text{ (from case 1)} + 24 \text{ (from case 2)} = 48 \]

### Final Answer:
The total number of numbers greater than 41000 that can be formed using the digits 1, 2, 3, 4, and 5 exactly once is **48**.