Question Video: Finding the Number of Ways to Arrange a Given Set of Digits to Form an 𝑛-Digit Number with Given Criteria Mathematics

Video Transcript

How many three-digit even numbers with no repeated digits can be formed using the elements of the set containing the numbers three, eight, nine, and two?

To calculate how many three-digit even numbers we can make, we’re going to use something called the product rule for counting. This will save us a little bit of time. Because rather than trying to list out all possible three-digit even numbers and hoping that we don’t lose any, it says that to find the total number of outcomes of two or more events, we simply multiply the number of outcomes of each event together.

We’re looking to create three-digit even numbers. Well, from this list, we can see that our number must end in either an eight or a two. So, we’ll begin by considering that digit. Since we know our number has to be even, there are two ways to choose its final digit. We can choose either a two or an eight.

And then, how many ways are there to choose the next digit in the even number? Well, since we’ve chosen either the two or the eight, we know that there are three numbers left to choose from. Then, once we’ve chosen the end digit and one of the other digits, there are two numbers left to choose from from our set. We’re only interested in three-digit even numbers, so we stop here.

And so, we calculate the number of three-digit even numbers we can make using the elements of our set by multiplying two by three by two, which is 12. There are 12 three-digit even numbers with no repeated digits that we can make using the elements of the set containing three, eight, nine, and two.