Here is my solution, using reasoning based on symmetry:
The total number of three digit numbers, greater than 800 with all three digits distinct, is 2 * 9 * 8 = 144.
We have two choices for the first digit (8 or 9), 9 choices for the second digit (it must be different from the first digit) and 8 choices for the third digit, as it should be different from the two previous digits).
Now, here is where, I think, symmetry can help:
Among those starting with 8, there are less even numbers, as the last digit cannot be 8 (so, only 4 choices), while odd choices for the last digit are 5.
For the numbers starting with 9, the situation is reversed, as there are only 4 choices for the third digit for odd numbers and 5 choices for the even numbers.
If we put all the numbers together, at the end, we have a balanced outcome, there must be the same number of each type.
It means that among the above 144 numbers, there are as many even as odd numbers, so 72 of each type.
Therefore, Answer C.
Did you meet questions where some type of reasoning based on symmetry can be used?
PhD in Applied Mathematics
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