Number of numbers greater than 24000 can be formed by using digits 1,2,3,4,5 when no digit being repeated is

Options

A. 36
B. 60
C. 84
D. 120

Answer

To solve the problem of finding how many numbers greater than 24000 can be formed using the digits 1, 2, 3, 4, and 5 without repetition, we can follow these steps:

### Step 1: Identify the range of numbers
We need to form numbers that are greater than 24000. This means we need to consider the first digit of the number carefully.

### Step 2: Determine the first digit
The first digit can be 2, 3, 4, or 5 to ensure the number is greater than 24000. If the first digit is 1, the number will definitely be less than 24000.

### Step 3: Count the possibilities based on the first digit
1. **If the first digit is 2**:
- The second digit can be 3, 4, or 5 (since we want numbers greater than 24000).
- If the second digit is 3, the remaining digits are 1, 4, 5 (3 choices for the last two digits).
- If the second digit is 4, the remaining digits are 1, 3, 5 (3 choices for the last two digits).
- If the second digit is 5, the remaining digits are 1, 3, 4 (3 choices for the last two digits).
- Total combinations when the first digit is 2: 3 (choices for second digit) × 3! (arrangements of remaining digits) = 3 × 6 = 18.

2. **If the first digit is 3**:
- The remaining digits can be any of 1, 2, 4, 5.
- Total combinations: 4! = 24.

3. **If the first digit is 4**:
- The remaining digits can be any of 1, 2, 3, 5.
- Total combinations: 4! = 24.

4. **If the first digit is 5**:
- The remaining digits can be any of 1, 2, 3, 4.
- Total combinations: 4! = 24.

### Step 4: Sum the possibilities
Now, we sum the total combinations from each case:
- From first digit 2: 18
- From first digit 3: 24
- From first digit 4: 24
- From first digit 5: 24

Total = 18 + 24 + 24 + 24 = 90.

### Final Answer
The total number of numbers greater than 24000 that can be formed using the digits 1, 2, 3, 4, and 5 without repetition is **90**.

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