In how many ways can a four-digit even number with no repeated digits be formed using the elements of the set containing the numbers one, six, nine, eight, seven, five?
To calculate the number of ways that we can create a four-digit even number, we don’t really want to use systematic listing. We don’t want to list all the possible options. Firstly, there could be quite a lot of them. And secondly, it’s really easy to lose some.
And so we recall the product rule for counting. This says that we can find the total number of outcomes for two or more events by multiplying the number of outcomes for each event together. In this case, our events are the digits we choose.
Now, we actually have six elements in our set. We’re only interested in calculating the number of four-digit numbers we can make. And so let’s just begin by considering what we mean for it to be an even number. Well, from our set, we know that the even numbers will all end in a six or an eight. So we’ll consider the last digit of our four-digit number first. And we say that, well, if the even numbers must end in a six or an eight, there are only two ways to choose the last digit of our even number. It can be six or eight. There are no other restrictions on our four-digit number.
And so once we’ve chosen the last digit, we’ve either chosen a six or an eight, we know that there are only five ways to choose the next digit. Since there are no repeated digits, this means that there are four ways to find the next digit in our number. And finally, once we’ve chosen the first three digits, we know that there are three elements left in our set. So there are three ways to choose the fourth digit.
The product rule says to find the total number of outcomes, we multiply the number of outcomes of each event together. So here that’s two times five times four times three. Well, two times five is 10, and four times three is 12. So we can calculate this quite quickly by working out 10 times 12, which is 120. This means there are 120 ways to choose a four-digit even number with no repeated digits from our set.