There are 900 three-digit integers. The number of three-digit integers having at least one repeated digit is also a three-digit number. If that number is represented by abc where each letter is a digit compute a-b+c.
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math - MathMate, Saturday, May 28, 2011 at 11:00pm
Here we have 10 digits, of which the first one cannot be zero.
If the three digits are distinct, there are 9 choices for the first digit, still 9 (including zero) for the second, and 8 for the third.
So 9*9*8=648 numbers.
There are therefore 900-648=252 numbers which have at least one repeated digit.
So a=2, b=5, c=2, and a-b+c=2-5+2=-1
To solve this problem, we need to find the number of three-digit integers that have at least one repeated digit.
First, let's calculate the total number of three-digit integers. A three-digit integer has a range from 100 to 999 inclusive. To determine the count, we subtract the lower limit from the upper limit and add 1:
Number of three-digit integers = 999 - 100 + 1 = 900.
Next, let's find the number of three-digit integers that do not have any repeated digits.
For a three-digit integer without any repeated digits:
- The hundreds digit can be any digit from 1 to 9, allowing 9 options.
- The tens digit can be any digit from 0 to 9, excluding the one already chosen for the hundreds digit. So, there are 9 options for the tens digit.
- The units digit can again be any digit from 0 to 9, excluding the two digits already chosen.
For the units digit, we have 8 options.
Using the fundamental counting principle, we multiply the number of options for each digit: 9 * 9 * 8 = 648.
The number of three-digit integers with at least one repeated digit is the complement of those without any repeated digits:
Number of three-digit integers with at least one repeated digit = Total number of three-digit integers - Number of three-digit integers without repeated digits
= 900 - 648
= 252.
Now that we know the number of three-digit integers with at least one repeated digit is 252, we can break it down into three digits: abc.
Here, a = 2, b = 5, and c = 2.
Finally, we calculate a - b + c:
a - b + c = 2 - 5 + 2
= -3 + 2
= -1.
Therefore, a - b + c = -1.