What is the greatest seven-digit number whose digits are all different?
Let’s split this problem into three quick steps. Firstly, let’s think about what a seven-digit number might look like. Then, how we can make the greatest seven-digit number. And then finally, the greatest seven-digit number whose digits are all different. So to start with, let’s think about seven-digit numbers. These are numbers that contain ones, tens, hundreds, thousands, tens of thousands, hundred thousands. And instead of thousand thousands, we call the next column millions. So we know that our seven-digit number is going to contain a number of millions.
Now, if we were asked to find the greatest seven-digit number, we just look to put the greatest digit in each of the places in our number. We know that the digits go from zero to nine. So nine is the greatest digit. You can’t get higher than nine. So we could put a nine in the millions place and nine in the hundred thousands place and so on, all the way down to nine ones. This would be the greatest seven-digit number we could make. And we’d read this as 9999999.
But we’re not asked to give the greatest seven-digit number. We’re asked to give the greatest seven-digit number whose digits are all different. So we’re going to need to change six of the digits in our number. We can only have one nine. And to think about where to put the digits in our number, we need to think about which place in our seven-digit number has the most importance. Which place has the most value? Well, this is the place furthest to the left. It’s the millions place. There’s no point putting a two or a five in the millions place when we can put a nine. So we’ll use our nine digit in the place where it has most value, nine million.
The place with the next greatest value is the hundred thousands place. But we can’t put a nine there. So we need to put the next best digit. Instead of a nine, we can put an eight. The place with the next greatest value is the ten thousands place. And again, we can’t put a nine there. And, of course, this time, we can’t use an eight either. We have to use the next greatest digit that we have available, which is the seven digit, seven lots of 10000 or 70000.
And we can see a pattern here. Nine, eight, seven. We just need to keep using the greatest possible digit that we have left each time. The digits in our number are in order from left to right, starting with the greatest digit and ending with the digit three. We’re going to end with three ones. We can write our seven-digit number using commas to separate the millions from the hundred thousands place and also to separate the thousands from the hundreds place. This helps us to read the number more clearly.
We’ve found the greatest seven-digit number whose digits are all different by thinking about the place value of each digit and putting the digits in order from largest to smallest so that the largest digits were in the places with the largest value. The greatest seven-digit number whose digits are all different is 9876543.