Video Transcript
How many three-digit numbers that are less than 900 and that have no repeated digits can be formed using the elements of the set seven, one, nine?
There are a couple of ways that we can find these three-digit numbers. Firstly, we could just list the possible numbers systematically. As the number must be less than 900, we know that the first digit, that in the hundreds column, cannot be nine. This means that the first digit must be either seven or one. Let’s begin with the smallest digit one. We place this in the hundreds column. And since our number can have no repeated digits, we must place a seven and a nine in the tens and units column. One way of doing this gives us the number 179. We could also place the nine and seven the other way round, giving us 197. These are the only two numbers we can create with a one in the hundreds column.
Repeating this process by placing seven in the hundreds column, we have the numbers 719 and 791. As already mentioned, we cannot place nine in the hundreds column. And we can therefore conclude that we can make four three-digit numbers, all of which are less than 900 with no repeated digits from the set containing the numbers seven, one, and nine.
An alternative method to solve this problem is to use the product rule for counting. This states that we can find the number of possible outcomes made by combining two or more events by multiplying the number of possible outcomes of each event together. In this case, our events are the numbers we choose; they are our three digits. Since our number has to be less than 900, its first digit must be a seven or a one. This means that there are two possible ways of choosing that first digit. Next, we consider the second digit. We’ve chosen one number from our set, meaning that we have two numbers left. So, there are two ways to choose a second digit to ensure that we don’t have any repeated digits.
Finally, we consider the third digit. As we’ve chosen two numbers from our set and we know our digits cannot repeat, there is only one way to choose that third digit. The product rule says that we can find the number of three-digit numbers we can make by multiplying these values together. That’s two multiplied by two multiplied by one, which once again gives us an answer of four. There are four three-digit numbers that are less than 900 and have no repeated digits that can be formed using the elements of the set seven, one, nine.
