In how many ways can a three-digit number, starting with an even digit and containing no repeated digits, be formed from the numbers one through eight?
Now what we’re not going to do is try and list out all possible three-digit numbers from our list. Firstly, there are quite a lot of numbers to choose from, so that could take us quite a long time. And secondly, it would be really easy to miss out a number. Instead, we’re going to use something called the product rule for counting. It tells us that to find the total number of outcomes for two or more events, we multiply the number of outcomes for each event together. It’s called the product rule because it involves multiplying to find a product.
Here, the events are the digits we choose. So, let’s begin by looking at the first digit. We’re told this needs to be an even digit. And so, from our list, it can be two, four, six, or eight. And so, we see that there are four ways of choosing the first digit. There are four outcomes for our first event. Now, we’re choosing from a list of eight numbers, and we want that to be no repeated digits. So once we’ve chosen our first digit, we know that there can only be seven left to choose from for our second digit.
Finally, we choose the third digit in our number. We’ve already chosen the first and second digits, meaning there must only be six numbers left in our list. The product rule says that to find the total number of outcomes, we multiply the number of outcomes of each event together. That’s four times seven times six, where seven times six is 42. And we can multiply 42 by four by doubling and then doubling again, giving us 168.
There are 168 ways to choose a three-digit number, starting with an even digit from the numbers one, two, three, four, five, six, seven, eight.