How many three digit can be formed using the digits 1, 2, 3, 4, 5, without repetition of digits? How many of these are even?

Answer

To solve the problem of how many three-digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition of digits, and how many of these are even, we can follow these steps:

### Step 1: Calculate the total number of three-digit numbers
1. **Choose the first digit**: We have 5 options (1, 2, 3, 4, 5).
2. **Choose the second digit**: After selecting the first digit, we have 4 remaining options.
3. **Choose the third digit**: After selecting the first and second digits, we have 3 remaining options.

The total number of three-digit numbers can be calculated as:
\[ \text{Total three-digit numbers} = 5 \times 4 \times 3 \]
Calculating this gives:
\[ 5 \times 4 = 20 \\ 20 \times 3 = 60 \]
So, the total number of three-digit numbers that can be formed is **60**.

### Step 2: Calculate the number of even three-digit numbers
A number is even if its last digit is an even number. In our case, the even digits available are 2 and 4.

**Case 1: Last digit is 2**
1. If 2 is the last digit, we have 4 remaining digits (1, 3, 4, 5) to choose from for the first two positions.
2. **Choose the first digit**: We have 4 options (1, 3, 4, 5).
3. **Choose the second digit**: After selecting the first digit, we have 3 remaining options.

The total number of three-digit numbers with 2 as the last digit is:
\[ \text{Total (last digit 2)} = 4 \times 3 = 12 \]

**Case 2: Last digit is 4**
1. If 4 is the last digit, we have 4 remaining digits (1, 2, 3, 5) to choose from for the first two positions.
2. **Choose the first digit**: We have 4 options (1, 2, 3, 5).
3. **Choose the second digit**: After selecting the first digit, we have 3 remaining options.

The total number of three-digit numbers with 4 as the last digit is:
\[ \text{Total (last digit 4)} = 4 \times 3 = 12 \]

### Step 3: Calculate the total number of even three-digit numbers
Now, we add the totals from both cases:
\[ \text{Total even three-digit numbers} = 12 + 12 = 24 \]

### Final Answer
- The total number of three-digit numbers that can be formed is **60**.
- The total number of even three-digit numbers is **24**.