To solve the problem of how many 5-digit and 3-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 without repetition, we can follow these steps:
### Step 1: Calculate the number of 5-digit numbers
1. **Identify the total digits available**: We have the digits 1, 2, 3, 4, and 5.
2. **Determine the number of positions**: A 5-digit number has 5 positions to fill.
3. **Fill the first position**: We can choose any of the 5 digits for the first position. So, there are 5 options.
4. **Fill the second position**: After using one digit, we have 4 digits left. Thus, there are 4 options for the second position.
5. **Fill the third position**: After using two digits, we have 3 digits left. Therefore, there are 3 options for the third position.
6. **Fill the fourth position**: After using three digits, we have 2 digits left. So, there are 2 options for the fourth position.
7. **Fill the fifth position**: Finally, after using four digits, we have 1 digit left. Thus, there is 1 option for the fifth position.
8. **Calculate the total number of 5-digit combinations**: Multiply the number of options for each position:
\[
5 \times 4 \times 3 \times 2 \times 1 = 120
\]
Therefore, the total number of 5-digit numbers that can be formed is **120**.
### Step 2: Calculate the number of 3-digit numbers
1. **Identify the total digits available**: We still have the digits 1, 2, 3, 4, and 5.
2. **Determine the number of positions**: A 3-digit number has 3 positions to fill.
3. **Fill the first position**: We can choose any of the 5 digits for the first position. So, there are 5 options.
4. **Fill the second position**: After using one digit, we have 4 digits left. Thus, there are 4 options for the second position.
5. **Fill the third position**: After using two digits, we have 3 digits left. Therefore, there are 3 options for the third position.
6. **Calculate the total number of 3-digit combinations**: Multiply the number of options for each position:
\[
5 \times 4 \times 3 = 60
\]
Therefore, the total number of 3-digit numbers that can be formed is **60**.
### Final Answer:
- (i) The number of 5-digit numbers that can be formed is **120**.
- (ii) The number of 3-digit numbers that can be formed is **60**.
