In how many ways can a three-digit number which is greater than 300 and has no repeated digits be formed using the numbers one, two, three, four, five, six?
We’re looking to make a three-digit number from six possible numbers, one through six. So, what we’re not going to do is list out all possible numbers. There might be quite a lot of them, and it could take us quite a long time. Instead, we use something called the product rule of counting. And this says that to find the total number of outcomes for two or more events, we multiply the number of outcomes for each event together.
In this case, each event is the digit we choose for our three-digit number. So, let’s begin by looking at the first digit. Now, our three-digit number needs to be greater than 300. And so, it can be three, four, five, or six. This means there are four ways of choosing the first digit for our number. We’ll now consider the second event. That is the number of ways we can choose the second digit.
Well, there are six digits altogether. We’ve already chosen one; we chose either three, four, five, and six. And we know that can be no repeated digits. So, that means we’re left with five possible digits to choose from. In the same way we consider the number of ways we can choose our third digit. Remember we started with six digits and we’ve already chosen two. We want there to be no repeats, so we’re left with the possible four digits to choose from.
The product rule says that we can find the total number of outcomes by multiplying these values together. That’s four times five times four, which is equal to 80. There are 80 ways to choose a three-digit number greater than 300 with no repeated digits using the numbers one through six.