What is the smallest five-digit number whose digits are all different?
In this problem, we need to make a five-digit number. And the first thing we can say about it is that all of its digits need to be different. Because we’re trying to make the smallest five-digit number, this means then that we need to use the smallest five digits. So that’s zero, one, two, three, and four. If we replace any of these five digits with any of the other digits, then our number isn’t going to be the smallest possible number anymore. To make the smallest possible number, we have to use these five digits.
A five-digit number means there are five different places where we could put our digits. From left to right, these have the value of 10 thousands. This is the place with the greatest possible value, thousands, hundreds, tens, and ones. This is the place with the lowest possible value. If we look at the way we’ve written our digits to start with, we can see these are in order, from smallest to largest. So surely, if we’d like to make the smallest five-digit number, we should write our digits in the same order from smallest to largest in our place value grid. This way, we put the lowest possible value in our 10 thousands column, the next lowest value in our thousands column, and so on.
Now, this isn’t the right answer. But it’s almost correct. If we try to read this number, it’s not actually a five-digit number. The zero in the 10 thousands column doesn’t stand for anything. So our number is actually 1234. It’s a four-digit number. For our number to be a true five-digit number, we need to put a digit that stands for something other than zero in our 10 thousands place. But remember, we do want it to be as small as possible. What if we swap around the zero and the one. We now have one lot of 10 thousand and zero thousands. And importantly, we now have a five-digit number. Our digits are in order from left to right from smallest to largest, with one exception. We’ve swapped the zero and the one around so that we actually make a five-digit number.
The smallest possible five-digit number, whose digits are all different, is 10234.