Question

1. For three-digit numbers, find how many numbers fall into each of the following categories.Using permutations and fractions. a. All digits are different 6. Both digits are the same s. All three digits are the same

Answer 1

For three-digit numbers, the number of numbers falling into each category of having different digits, two same digits, and all three same digits can be calculated using permutations.

Step-by-step explanation:

1. For the category of three-digit numbers where all digits are different, we can use the concept of permutations. The first digit can be any number from 1 to 9, so there are 9 choices. The second digit can be any number from 0 to 9 (excluding the digit chosen for the first digit), so there are 9 choices. And the third digit can be any number from 0 to 9 (excluding the two digits chosen for the first and second digits), so there are 8 choices. Therefore, the total number of three-digit numbers with all digits different is 9 x 9 x 8 = 648.
2. For the category of three-digit numbers where both digits are the same, we can choose the repeating digit in 9 ways (from 1 to 9) and then select any of the remaining 9 digits for the third digit. So, the total number of three-digit numbers with two digits being the same is 9 x 9 = 81.
3. For the category of three-digit numbers where all three digits are the same, there is only 1 possibility, which is the number with all three digits as the chosen digit. So, the total number of three-digit numbers with all three digits being the same is 1.

Learn more about Counting three-digit numbers with different digits, two same digits, and all three same digits