How many 6-digit numbers can be formed form the digits 0, 1, 3, 5, 7, 9 when no digit is repeated? How many of them are divisible by 10?

Explanation

To solve the problem, we need to find how many 6-digit numbers can be formed using the digits 0, 1, 3, 5, 7, and 9 without repeating any digits, and then determine how many of those numbers are divisible by 10.

### Step 1: Calculate Total 6-Digit Numbers
1. **Identify the digits**: The available digits are 0, 1, 3, 5, 7, and 9.
2. **First digit selection**: The first digit of a 6-digit number cannot be 0 (as it would not be a 6-digit number). Therefore, we can choose from the digits 1, 3, 5, 7, or 9. This gives us **5 options** for the first digit.
3. **Subsequent digit selections**: After choosing the first digit, we have 5 digits remaining (including 0) to fill the next positions:
- For the second digit, we have **5 options** (since we can now include 0).
- For the third digit, we have **4 options** (one digit has been used).
- For the fourth digit, we have **3 options**.
- For the fifth digit, we have **2 options**.
- For the sixth digit, we have **1 option**.

4. **Calculate the total combinations**:
\[ \text{Total combinations} = 5 \times 5 \times 4 \times 3 \times 2 \times 1 \]
\[ = 5 \times 120 = 600 \]

### Step 2: Calculate 6-Digit Numbers Divisible by 10
1. **Divisibility by 10 condition**: A number is divisible by 10 if its last digit is 0.
2. **Last digit selection**: Since the last digit must be 0, we only have **1 option** for the last digit.
3. **First digit selection**: The first digit can be any of the remaining digits (1, 3, 5, 7, or 9), giving us **5 options**.
4. **Subsequent digit selections**: After choosing the first digit, we have 4 digits left (including 0) to fill the next positions:
- For the second digit, we have **4 options**.
- For the third digit, we have **3 options**.
- For the fourth digit, we have **2 options**.
- For the fifth digit, we have **1 option**.

5. **Calculate the total combinations for numbers divisible by 10**:
\[ \text{Total combinations divisible by 10} = 5 \times 4 \times 3 \times 2 \times 1 \]
\[ = 5 \times 24 = 120 \]

### Final Answers
- The total number of 6-digit numbers that can be formed is **600**.
- The total number of 6-digit numbers that are divisible by 10 is **120**.