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
(I came up with 252 integers so when I did the computation it didn't make sense)
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 find the number of three-digit integers having at least one repeated digit, we can use the principle of inclusion-exclusion.
First, let's determine the total number of three-digit integers. A three-digit integer has a hundreds place digit (from 1 to 9), a tens place digit (from 0 to 9), and a units place digit (from 0 to 9). So, there are 9 choices for the hundreds place digit, 10 choices for the tens place digit, and 10 choices for the units place digit. This gives us a total of 9 * 10 * 10 = 900 three-digit integers.
Now, let's calculate the number of three-digit integers without any repeated digits. To do this, we consider the choices for each place value. Since the hundreds digit cannot be 0, there are 9 choices for that position. Then, for the tens digit, there are 9 choices (excluding the digit already chosen for the hundreds place). Finally, for the units digit, there are 8 choices (excluding both the hundreds and tens digits chosen earlier). Therefore, the number of three-digit integers without any repeated digits is 9 * 9 * 8 = 648.
Finally, to find the number of three-digit integers having at least one repeated digit, we subtract the number of three-digit integers without any repeated digits from the total number of three-digit integers: 900 - 648 = 252.
Now, we can represent this number as abc, where a = 2, b = 5, and c = 2.
To compute a - b + c, we substitute the values: 2 - 5 + 2 = -1 + 2 = 1.
Therefore, a - b + c equals 1.