To address the given questions, we need to determine two things:
1. The number of 6-digit numbers that can be formed using the digits 0-9 without repetition:
2. How many of these numbers are divisible by 10:
### Part 1: Number of 6-Digit Numbers
When forming a 6-digit number using the digits 0 to 9 without repetition, several considerations must be made:
- No repetition of digits: A 6-digit number means exactly 6 different digits must be used out of the 10 available digits (0-9).
- First digit cannot be 0: For a number to be a valid 6-digit number, the first digit cannot be 0 (otherwise it would be a 5-digit number). Therefore, there are 9 choices for the first digit (1-9).
Let's break it down step-by-step:
1. Choose the first digit (cannot be 0): We have 9 choices (1-9).
2. Choose the remaining 5 digits from the remaining 9 digits (including 0):
- For the second digit, we now have 9 options remaining.
- For the third digit, we have 8 options left, and so on down to 5 options for the sixth digit.
The number of ways to arrange these 6 digits where no digit is repeated and the first digit is not 0 is calculated as:
[tex]\[ 9 \times 9! / 4! \][/tex]
### Part 2: Number of 6-Digit Numbers Divisible by 10
For a number to be divisible by 10, the last digit must be 0. Ensuring that there is no repetition of digits, and considering 0 can no longer be a choice for any other position:
1. Last digit is fixed as 0.
2. Choose the first digit (cannot be 0): We have 9 choices (1-9).
3. Choose the remaining 4 digits from the remaining 8 digits:
- For the second digit, we have 8 options left.
- For the third digit, we have 7, and so on.
The number of 6-digit numbers divisible by 10 and without repetition of digits can be computed as:
[tex]\[ 9 \times 8! / 3! \][/tex]
### Final Answer
Following these steps carefully, we get the results as:
1. The number of 6-digit numbers that can be formed using the digits 0 to 9 without repetition: 136,080
2. The number of 6-digit numbers divisible by 10: 3,024
