How many numbers can be formed from 1, 2, 3, 4, 5 (without repetition), when the digit at the unit's place must be greater than that in the ten's place?

Options

A. 54
B. 60
C. `(5!)/(3)`
D. `2 xx 4!`

Answer

To solve the problem of how many numbers can be formed from the digits 1, 2, 3, 4, 5 (without repetition) such that the digit at the unit's place is greater than the digit at the ten's place, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem:**
We need to form numbers using the digits 1, 2, 3, 4, and 5. The condition is that the digit in the unit's place must be greater than the digit in the ten's place.

2. **Choosing the Ten's and Unit's Places:**
We can denote the ten's place as \( T \) and the unit's place as \( U \). The condition we need to satisfy is \( U > T \).

3. **Possible Combinations:**
Let's analyze the possible pairs of \( (T, U) \):
- If \( T = 1 \), then \( U \) can be 2, 3, 4, or 5 (4 options).
- If \( T = 2 \), then \( U \) can be 3, 4, or 5 (3 options).
- If \( T = 3 \), then \( U \) can be 4 or 5 (2 options).
- If \( T = 4 \), then \( U \) can only be 5 (1 option).
- If \( T = 5 \), then there are no valid \( U \) (0 options).

4. **Counting Valid Pairs:**
Now we can count the total valid pairs:
- For \( T = 1 \): 4 options (2, 3, 4, 5)
- For \( T = 2 \): 3 options (3, 4, 5)
- For \( T = 3 \): 2 options (4, 5)
- For \( T = 4 \): 1 option (5)
- For \( T = 5 \): 0 options

Total valid pairs = \( 4 + 3 + 2 + 1 + 0 = 10 \).

5. **Choosing Remaining Digits:**
After choosing \( T \) and \( U \), we have 3 remaining digits from which we can choose to fill the remaining places (hundreds, thousands, etc.). The number of ways to arrange these 3 digits is \( 3! = 6 \).

6. **Calculating Total Numbers:**
Therefore, the total number of valid numbers is given by:
\[ \text{Total numbers} = \text{(number of valid pairs)} \times \text{(arrangements of remaining digits)} = 10 \times 6 = 60. \]

### Final Answer:
The total number of numbers that can be formed is **60**.