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

How many two-digit numbers, which end with the digit 2 and have no repeated digits, can be formed using the elements of the set {3, 1, 2}?

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Video Transcript

How many two-digit numbers, which end with the digit two and have no repeated digits, can be formed using the elements of the set three, one, two?

There are two ways we can answer this question. The first is called systematic listing. Since we have a very small set of numbers from which to choose from, we know the outcomes for an event, that is, the number of two-digit numbers we can make, can be listed or organized in a systematic way. So, let’s see what that would look like. We need our number to end with the digit two. And so, this means its first digit. Remember, we’re only interested in a two-digit number. And the digits aren’t repeated can either be three or one. And so, the numbers that we can make are either 32 or 12. There are two ways to make two-digit numbers which end with the digit two and have no repeated digit from our set.

Now, an alternative method we could’ve used is the product rule for counting. This can be nicer when we’re dealing with a larger number of events. It 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, our events are the numbers we choose. So, we begin by looking at our restriction: our two-digit number must end with the digit two. And so, there’s actually only one way of choosing the second digit from the elements of our set.

Then, we look at the other digit in our two-digit number. Remember, there are no repeated digits and we have three numbers in our set. This means once we’ve chosen that second digit, there are only two numbers left to choose from for the first digit. The product rule says that we need to multiply these to find the total number of outcomes. That’s one times two, which once again gives us two.

There are two two-digit numbers which end with the digit two that can be made using elements of the set containing the numbers three, one, and two.

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