Combinatorial Analysis: The Fundamental Principle of Counting

Question

To solve the following problem: "How many natural numbers with three distinct digits are there?". The best way would be to analyze how many different ways I can choose the first digit (there are nine, because I couldn't choose zero). For the second, there are nine too (because I could choose zero, but I couldn't choose the number I have already chosen for the first case ). For the third digit it would be eight, (because I cannot repeat either the first or the second one). If I started solving the problem by the last digit, the ways I could choose the first would depend (if the last digit was zero, I could choose the first one of nine different ways and if it was non-zero, I could choose the first only eight different ways). So this way is more difficult to solve the problem. Now let's imagine that I have the following problem: "How many even natural numbers of three distinct digits are there?". Here, I can no longer start the following choice: "For the first digit there are nine different ways (since zero cannot be chosen), for the second there are also nine, and for the third there are two (2,4,6, 8, but since one of them has already been chosen for the first and one for the second, there are two left.) I know this is wrong, but why? How do I know in a counting problem which decision to start with?

Answer 1

"And for the third there are ____" there are some number between three and five options remaining from the even digits... In order to apply rule of product (multiplication principle) you must have the amount of options be unambiguous and not relying on what was previously selected at earlier steps (though the list of options may vary). "How do I know which decision to start with" Practice and intuition... as a general rule of thumb, try picking whatever is the most restrictive first... though this is merely a rule of thumb and not always guaranteed to work.

For a correct approach, pick the units digit first. Then... break into cases based on whether the units digit selected was a zero or not. This gives $1\cdot 9\cdot 8 + 4\cdot 8\cdot 8=328$ outcomes. Alternatively, ignore that leading zeroes aren't allowed for your initial count and then correct the count after the fact for a total of $5\cdot 9\cdot 8 - 1\cdot 4\cdot 8=328$ (for the first pass, pick units digit, then hundreds digit and then tens... for the second pass pick hundreds digit, then units digit, then tens)