What is the largest nine-digit number that can be formed using all the digits of six, six, seven, nine, seven, five, zero, one, and three?
As part of the question, we’re given nine digits. Those nine digits we’ve just read out. And we need to rearrange those digits so that they make the largest possible nine-digit number. Now, we know that the value of a digit in a number depends on where we write it. For example, the digit three might have a value of three tens or three thousands, maybe even three millions. It depends where we write it in the number.
So, perhaps the first thing we should do is to draw a place value grid to think about what the value of each digit in a nine-digit number is going to be. We can think of our digits in groups of three. So, going from right to left, we could call our first group of three the ones. And we have the ones, tens, and hundreds digits. Our second group of three are our thousands. Here, we have the thousands digit, ten thousands digit, and hundred thousands digit. Our final group of three are our millions. We have the millions, tens of millions, and hundred millions digits.
Let’s start by thinking about the hundred millions place. If we want to make a digit that’s the largest possible nine-digit number, which digit do we want to put in the hundred millions place? This is the place with the largest value. Well, there’s no point putting a one in there because there are other digits that are greater. We could even do better than a three or a five. Instead of just going through each digit, we can say to ourselves that we just need to pick the largest possible digit.
And if we look at our list of digits, the largest possible digit is nine. So, the largest possible number will have nine hundred millions. It can sometimes be tricky to spot which digits we’ve used and which we haven’t. So, what we’ll do is we’ll write them out underneath and we’ll rub them out as we use each one. So, to begin with, we’ve used the digit nine.
The next place with the greatest value is the ten millions place. And again, to make this number have the largest possible value, we need to pick the next largest digit. We’ve used the digit nine, so the digit with the next largest value is seven. So, let’s cross off seven from our list. And there’s another seven in our list, so we can put this in the millions place. What we’re doing really is putting the digits in order from largest to smallest. This is going to give us the largest possible nine-digit number.
The next largest digits are two sixes. And we can put these in the hundred thousands and ten thousands places. In our thousands place, we need to put the digit five. And then, we’re left with a digit zero, one, and three. And if we put these in order from largest to smallest, we get 310.
We’ve found the answer by rearranging the digits from largest to smallest so that the largest digit has the largest possible value and the smallest digit has the smallest possible value. So, the largest nine-digit number that we can make using the digits six, six, seven, nine, seven, five, zero, one, three is 977665310.