how many 3-digit numbers have exactly 2 digits that are the same?
Answer is 243
Starting at 100 (the first 3-digit number), I counted all numbers that had exactly two digits that were the same. I did this through 199. There are 27 of these. I then multiplied that result by 9 (to pick up the 200s through 900s).
got that from google :)
Starting at 100 (the first 3-digit number), I counted all numbers that had exactly two digits that were the same. I did this through 199. There are 27 of these. I then multiplied that result by 9 (to pick up the 200s through 900s).
To find the number of 3-digit numbers that have exactly 2 digits that are the same, we can consider the different possibilities for the repeating digit.
1. The repeating digit is the first digit:
- In this case, the first digit can be any of the 1 to 9 digits (since it cannot be zero).
- The second digit can be any of the 10 digits (0 to 9) excluding the repeated digit.
- The third digit can be any of the 10 digits (0 to 9) excluding both the repeated digit and the second digit.
- Therefore, there are (9 choices for the first digit) × (9 choices for the second digit) × (8 choices for the third digit) = 648 numbers in this case.
2. The repeating digit is the second digit:
- In this case, the first digit can be any of the 9 non-zero digits.
- The second digit can be any of the 9 digits excluding the first digit.
- The third digit can be any of the 10 digits excluding both the repeating digit and the first digit.
- Therefore, there are (9 choices for the first digit) × (9 choices for the second digit) × (8 choices for the third digit) = 648 numbers in this case.
3. The repeating digit is the third digit:
- In this case, the first digit can be any of the 9 non-zero digits.
- The second digit can be any of the 10 digits excluding the repeating digit.
- The third digit can be any of the 9 digits excluding the first digit.
- Therefore, there are (9 choices for the first digit) × (10 choices for the second digit) × (9 choices for the third digit) = 810 numbers in this case.
Finally, to find the total number of 3-digit numbers with exactly 2 digits that are the same, we add up the results from each case:
648 + 648 + 810 = 2106.
Therefore, there are 2106 3-digit numbers that have exactly 2 digits that are the same.
To find out how many 3-digit numbers have exactly 2 digits that are the same, we can break down the problem into cases.
Case 1: The two repeated digits are the first and second digits.
In this case, we have 9 choices for the first digit (excluding 0), and 10 choices for the third digit (including 0). The digit that is repeated can be any of the 9 possible digits (excluding 0 and the digit chosen for the first digit). Thus, there are 9 * 1 * 10 = 90 numbers in this case.
Case 2: The two repeated digits are the second and third digits.
Similar to Case 1, we have 10 choices for the first digit, 9 choices for the second digit (excluding the digit chosen for the first digit), and 1 choice for the third digit. So, there are 10 * 9 * 1 = 90 numbers in this case.
Case 3: The two repeated digits are the first and third digits.
In this case, we have 9 choices for the first digit (excluding 0), 10 choices for the second digit (including 0), and 1 choice for the third digit. So, there are 9 * 10 * 1 = 90 numbers in this case.
To find the total number of 3-digit numbers with exactly 2 repeated digits, we sum up the results from each case: 90 + 90 + 90 = 270.
Therefore, there are 270 3-digit numbers that have exactly 2 digits that are the same.